In a nonstick skillet cook onion in oil over medium-high heat until soft, about 5 minutes.
Reduce heat to medium-low, add rosemary and cook, covered, until onion is deep brown and caramelized, about 20 minutes. Stir in salt, transfer onion mixture to a bowl and cool.
Meanwhile toast walnuts on a baking sheet at 350° until just golden, 3-5 minutes. Remove, cool, roughly chop and set aside.
Place phyllo cups on a baking sheet and cook until toasted and golden, 3-5 minutes.
In the bowl of a stand mixer, beat together chèvre, Chobani, onion mixture and toasted walnuts. Divide filling among phyllo cups and serve.