Melt 2 tablespoons butter in a large skillet over medium-high heat.
Add the chopped onion and chopped green bell pepper. Cook, stirring occasionally, until they begin to soften but still have a little crunch, about 3–4 minutes.
Add the finely chopped chipotle peppers in adobo, crushed red pepper flakes, smoked paprika, and season to taste with salt and pepper. Stir and cook about 1 minute to bloom the spices.
Stir in the shredded leftover beef short ribs or brisket and cook until heated through, about 2–3 minutes. Transfer the beef/onion mixture to a bowl.
Wipe the skillet clean and return it to medium heat.
Divide the beef/onion mixture into 4 equal portions. To assemble each quesadilla: place one 8-inch flour tortilla on your work surface, sprinkle about 1/4 cup shredded Mexican cheese over it, top with 1 portion (1/4) of the beef mixture, sprinkle another 1/4 cup cheese over the beef, then place a second tortilla on top.
Add about 1/2 tablespoon butter to the preheated skillet and swirl to coat. Place one assembled quesadilla in the skillet and cook until the bottom is golden brown and the cheese starts to melt, about 2–3 minutes.
Carefully flip the quesadilla and cook the other side until golden brown and the cheese is fully melted, about 2–3 minutes more. Press gently with a spatula if desired.
Transfer the cooked quesadilla to a cutting board, let rest 1 minute, then cut into wedges. Repeat steps 7–9 for the remaining 3 quesadillas, adding about 1/2 tablespoon butter to the skillet for each one.