Preheat the oven to 200 degrees F and arrange a baking sheet in the oven.
Meanwhile, in a large nonstick skillet over medium heat, warm the oil. Add the bell peppers and cook, stirring occasionally, until browned and very tender, 8 to 10 minutes. Transfer the peppers to a plate or bowl and set aside. Carefully wipe out the skillet and set aside.
Lightly spread the butter on one side of each tortilla. Lay the tortillas, buttered side down, on a work surface. Arrange 1/2 cup of cheese on half of each tortilla, leaving a 1-inch border at the edge. Arrange the bell peppers and pork on top, then fold the tortillas in half over the fillings.
Rewarm the skillet over medium heat. Add 2 of the quesadillas and cook, covered, until golden brown on the bottom, 2 to 4 minutes. Carefully turn and cook, uncovered, until golden brown on the other bottom half and the cheese has melted, 1 to 3 minutes.
Transfer the finished quesadillas to the baking sheet in the oven to keep warm. Carefully wipe out the skillet and repeat with the remaining quesadillas. Cut each quesadilla into 4 to 6 wedges. Serve with salsa, if using.