We have already seen in a previous post that the ancient historian Plutarch, a Platonist philosopher and initiated priest of Delphi, explained why the priests of Isis abstained from eating the flesh of various animals, as well abstaining from wearing clothing made from animal matter including wool.  

The Plutarch passage discussed previously came from his discourse on Isis and Osiris.  However, Plutarch also wrote two other more thorough discussions of the question of abstaining from the eating of meat, entitled De Esu Carnium, ("On the Eating of Flesh").  Both are fragmentary, meaning that the complete text as composed by the author has been lost to history, but what remains contains some rather forceful argument against the consumption of the meat of animals.

Here is a link to the first discourse: Plutarch,  On the Eating of Flesh, I.

Here is a link to the second: Plutarch,  On the Eating of Flesh, II.

The discourses begin with a series of questions from the author loaded with graphic language and packing quite a punch:

Can you really ask what reason Pythagoras had for abstaining from flesh? For my part I rather wonder both by what accident and in what state of soul or mind the first man who did so, touched his mouth to gore and brought his lips to the flesh of a dead creature, he who set forth tables of dead, stale bodies and ventured to call food and nourishment the parts that had a little before bellowed and cried, moved and lived. How could his eyes endure the slaughter when throats were slit and hides flayed and limbs torn from limb? How could his nose endure the stench? How was it that the pollution did not turn away his taste, which made contact with the sores of others and sucked juices and serums from mortal wounds? 

Later, Plutarch turns his rhetoric up another notch by imagining the reproach that the animals slaughtered for food might offer, if they "could recover feelings and voice," telling those who killed them for food that it was all unnecessary:

Oh blessed and beloved of the gods, you who live now, what an age has fallen to your lot wherein you enjoy and assimilate a heritage abounding in good things! How many plants grow for you! What vintages you gather! What wealth you may draw from the plains and what pleasant sustenance from trees! Why, you may even live luxuriously without the stain of blood.  [. . .] But you who live now, what madness, what frenzy drives you to the pollution of shedding blood, you who have such a superfluity of necessities? Why slander the earth by implying that she cannot support you? Why impiously offend law-giving Demeter and bring shame upon Dionysus, lord of the cultivated vine, the gracious one, as if you did not receive enough from their hands? Are you not ashamed to mingle domestic crops with blood and gore? You call serpents and panthers and lions savage, but you yourselves, by your own foul slaughters, leave them no room to outdo you in cruelty; for their slaughter is their living, yours is a mere appetizer.

In the second discourse on the subject, Plutarch brings up the doctrine of reincarnation, noting other philosophers who make this their main reason for avoiding the consumption of animals for food.  He does not go that far, he says, saying that there is room for doubt about whether souls do in fact "migrate from body to body."  However, he says that because there is enough doubt on either side of the reincarnation question, we should abstain from eating animals just in case, just as a soldier who is unsure whether or not a half-seen figure is friend or foe should err on the side of caution rather than risk killing a friend, saying:

Yet even if the argument of the migration of souls from body to body is not demonstrated to the point of complete belief, there is enough doubt to make us quite cautious and fearful. It is as though in a clash of armies by night you had drawn your sword and were rushing at a man whose fallen body was hidden by his armour and should hear someone remarking that he wasn't quite sure, but that he thought and believed that the prostrate figure was that of your son or brother or father or tent-mate — which would be the better course: to approve a false suspicion and spare your enemy as your friend, or to disregard an uncertain authority and kill your friend as your foe? The latter course you will declare to be shocking. 

This argument may strike modern readers as one that they can safely ignore, especially if they believe that "souls" do not really exist, or cannot survive the death of the body.  However, Chris Carter's excellent and important book Science and the Afterlife Experience contains reports of rather rigorous modern examinations of the question of reincarnation, and some evidence that the possibility should not be hastily dismissed.  

In that book, Chris Carter notes the widespread belief in reincarnation outside of cultures that have historically been heavily influenced by orthodox Christianity, Judaism, and Islam, and also points out that:

at least some Christians believed in reincarnation up until the sixth century.  Although it was not part of official instruction, leaders of the church appear to have tolerated the belief as acceptable, until the Council of Nice in 553 CE. 

and in a footnote to this discussion, Chris Carter points out the following:

there are at least two references to reincarnation in the New Testament.  At one point the disciples ask Jesus if a blind man sinned in a previous life, and Jesus did not rebuke them (John 9:1-2); at another point Jesus describes John the Baptist as the prophet Elijah reborn (Matthew 11:11-15).  18-19 and footnote on 19.

However, it is likely that the disappearance of the doctrine of reincarnation discussed above is connected to the historic decline in vegetarian practice in some parts of the world, as well as the continuation of the practice of vegetarianism among at least some parts of the population in parts of the world that continued to believe in reincarnation (including areas to the east of the lands conquered by the Roman Empire, such as India, Tibet, and China).  

The fact that vegetarianism clearly had some very strong advocates in the west in ancient times and that the practice continued in other parts of the world (such as the lands to the east) may be an important clue, and it may tell us that the most ancient cultures around the world, including apparently the priests of ancient Egypt, taught abstinence from eating meat.  

As Plutarch says at the beginning of his discourses (and I paraphrase), the question might not be "when did vegetarianism begin?" but rather, "When and why was that teaching discarded, and the eating of meat initiated?"