Add 2 tablespoons olive oil to a large cast iron or non-stick skillet over high heat.
Add in the sliced bell pepper and cook until tender and dark brown/black areas appear, about 3 to 4 minutes. Remove and set aside.
Sizzle 1/2 tablespoon of the butter in a cast iron or non-stick skillet over medium heat and lay a flour tortilla in the skillet.
Build the quesadillas by laying grated cheese onto the tortilla, and then arranging the chicken, cooked pepper and jalapeno rounds. Top with a little more grated cheese and top with a second tortilla.
When the tortilla is golden on the bottom side, carefully flip the quesadilla to the other side, adding another 1/2 tablespoon butter to the skillet at the same time. Continue cooking until the second side is golden.
Repeat with the remaining tortillas and fillings.
Cut each quesadilla into wedges and serve with Pico de Gallo, sour cream and black beans on the side.