Heat a large non-stick skillet over medium/medium-high heat. Add 1 teaspoon oil, and swirl around in pan.
Add in mushrooms and sauté for about 5 minutes, until they release their juices. Stir in kale and sauté until it wilts, about 2 minutes. Transfer mushrooms & kale to a large mixing bowl.
Add another 1 teaspoon of oil to pan, and add onions & bell peppers. Sauté until softened, about 7 minutes. Transfer to bowl with the mushrooms & kale.
Stir all the vegetables together and add in goat cheese & butternut squash. {Adding the cheese while the vegetables are warm helps the cheese mix in completely}
Season with the red pepper flakes, parsley, nutmeg, paprika, and salt. Refrigerate for 30 minutes.
When ready to assemble, preheat the oven to 350. Spray a large baking sheet with non-stick spray. Take out 6 sheets of fillo dough. Wrap any unused portion of dough and refrigerate or freeze for another use.
Put the 6 sheets dough on a large working surface. Cut sheets in half. Cut 4 of those half-sheets in half again. You should have 8 half-sheets and 8 quarter-sheets.
Place one half sheet on the work surface. Put one quarter sheet on top of the half sheet. Fill with ⅓ cup vegetable-cheese mixture. . Fold the sides in & the bottom up. Then fold in a triangle shape {sort of like rolling up a flag}. --see below for step-by-step assembling pictures.
Brush or spray with remaining oil, and transfer to the prepared baking sheet. Repeat with remaining fillo sheets and vegetables-cheese mixture.
Bake in preheated oven for 20-25 minutes, until lightly browned & flaky.