Call fingers angled like branches what peel and cut apples,
to give to a girl who eats them in silence, looking.

"Call fingers angled like branches that peel and cut apples" is different. It means that the things cutting and peeling apples are the branches.

"What" makes the words that follow a noun phrase: "Call fingers the things that peel and cut apples"

So there's an important difference between "what" and "that" -- the difference between branches and fingers.