Heat 1 tablespoon of the butter and the olive oil in a large skillet over high heat. Add the squash and sprinkle with the salt, chile powder and pepper. Cook for several minutes, turning gently with a spatula, until the squash is deep golden brown and tender (but not falling apart). Remove to a plate and set aside.
In the same skillet, melt the remaining 1 tablespoon butter over medium-high heat and add in the kale. Toss it around with tongs and cook it for 3 to 4 minutes. Add in the cooked squash and gently toss together.
In a separate skillet, melt additional butter and lightly brown both sides of the tortillas.
Build the quesadillas one by one starting with a tortilla, adding a layer of the cheeses and a layer of the squash/kale mixture and ending with a second tortilla. You'll have four quesadillas when you're done.
Brown each quesadilla on both sides, adding more butter to the pan to make sure the tortillas are nice and golden and yummy. When the cheese is melted, remove each quesadilla from the skillet and cut it into four wedges.