In a medium mixing bowl, combine the BBQ sauce, honey, red pepper flakes, and cayenne pepper. Stir until the honey is fully integrated into the sauce.
Add the shredded chicken to the bowl and toss thoroughly until every strand is coated in the hot honey BBQ glaze.
Lay a flour tortilla flat and distribute 1/4 cup of shredded Monterey Jack cheese over one half of the surface.
Layer 1/2 cup of the glazed chicken and 1 tablespoon of scallions over the cheese layer.
Add another 1/4 cup of cheese over the chicken to act as a binder, then fold the tortilla in half.
Heat a heavy-bottomed skillet over medium heat and melt 1/4 of the butter.
Place the folded quesadilla in the skillet and toast for 3 to 4 minutes per side until the exterior is golden brown and the internal cheese is fully melted.
Remove from heat, let rest for 60 seconds to set the cheese, and slice into three wedges per tortilla.