Preheat oven to 350 degrees.
Place the butternut squash on the cutting board and use the vegetable peeler to remove the skin until a bright orange flesh is revealed. Use the knife to cut the squash in half longways. Use the spoon to remove the seeds from the squash. Then, cut the squash into small to medium size cubes. If you struggle cutting the squash, place it in the microwave for a couple minutes.
Heat 1 tbsp of butter and 1 tbsp oil in an ovenproof large skillet over high heat. Add the squash and season with a sprinkle of salt, pepper, and chili powder (optional). Cook for several minutes, turning with a spatula until the squash is lightly browned. Transfer skillet to a 350 degree oven and let cook until squash is tender. Remove from oven. Let cool slightly and crush ever so lightly with a fork. *** If you do not have an ovenproof skillet, place the squash on a baking sheet with 2 tbsp of oil and a sprinkle of salt, pepper, and chili powder (optional). Cook in oven until the squash is tender. Then follow the remaining instructions.
When squash has finished cooking, place one tortilla down and add shredded cheese on one half of the tortilla. Then add some of the cooked/crushed squash and a little more shredded cheese. Fold the tortilla in half. Repeat with remaining tortillas, squash, and cheese.
Once you have created all of the quesadillas, heat a large skillet on the stove over medium heat. Add a little butter or oil to the pan. Spread the butter and oil evenly over the bottom of the pan. Add one quesadilla and cook until brown on one side. Flip the quesadilla over and repeat until the quesadilla is brown on both sides and the cheese is melted. Repeat this step for each quesadilla.
When all quesadillas have finished, cut each one in half to serve. Enjoy with sour cream and/or your favorite salsa!