To make the pico de gallo, add all the ingredients to a small bowl and stir to combine. Set aside until ready to use.
Place peppers directly under oven broiler. When peppers begin to blister, rotate. Continue until peppers are blistered and charred on all sides – about 10 minutes. Remove from oven and place directly into a plastic bag and seal. After about 10 minutes, peel the peppers and cut them open to remove the seeds, then thinly slice.
To assemble the quesadillas, divide the cheese and peppers among 2 tortillas and top with the 2 remaining tortillas. Heat a large skillet over medium heat and add half the butter. When the butter has melted, add the quesadillas and cook for 2 to 3 minutes per side. Remove from heat.
Slice quesadillas into wedges and serve with pico de gallo.