Heat a nonstick skillet over medium heat and melt about half of the butter to coat the pan.
Place one tortilla in the skillet and sprinkle about one-third of the cheddar evenly over half the tortilla.
Top the cheese with about one-third of the shredded chicken, a few mango slices, a drizzle of BBQ sauce, and a sprinkle of chopped cilantro.
Fold the tortilla over the filling or place a second tortilla on top, press gently, and cook until the bottom is golden brown, about 2–3 minutes.
Carefully flip the quesadilla and cook the other side until golden and the cheese is melted, another 2–3 minutes. Add more butter to the pan if needed.
Repeat with the remaining tortillas and filling (about two more quesadillas), using the remaining butter as needed.
Transfer finished quesadillas to a cutting board, cut into triangles, and serve warm.