Which enzyme begins starch digestion in the mouth?

Amylase, in saliva, which is why bread starts tasting sweet if you hold it in your mouth long enough - the enzyme is cutting long starch chains into sugars while you sit there.

How much of it you make is genetically variable and interesting. Humans carry multiple copies of the amylase gene, and the number differs between people and between populations - groups with a long history of starch-heavy agriculture tend to carry more copies than traditionally hunting or herding populations. It is one of the clearer examples of recent human evolution tracking diet, and dogs show a parallel change relative to wolves, from living alongside farmers.

The mouth stage does not last long. Stomach acid destroys salivary amylase within minutes, and most starch digestion happens later in the small intestine with pancreatic amylase. What the mouth stage is really doing is starting the process and telling the rest of the system what is coming - which is also why taste evolved to find starch and sugar rewarding, in a world where they were scarce rather than engineered into everything.