In a STB. When the lye breaks down the plant material, does the biosynthesis of L-tryptophan begin then or is it already done within the plant itself?

??

you denature the proteins when basifying.
there is no biosynthesis in extraction chemistry.

for sake of answering the question, live plants produce their own tryptophan via
the shikimic acid pathway.

Which proteins?

I'm trying to figure out the chemistry behind this.

I take it I'm asking about a more lab oriented approach vs the kitchen.

proteins in general.
when basifying, you exceed physiological pH, disrupting protein function.