In a mixing bowl, combine the shredded chicken, barbecue sauce, and hot honey. Stir until the chicken is fully coated and flavorful.
Preheat the Blackstone griddle or a large skillet over medium heat. Lightly grease the surface with butter or oil.
Lay the tortillas on a clean surface. Sprinkle a layer of cheddar and Monterey Jack cheese over one half of each tortilla.
Spoon the BBQ chicken mixture evenly over the cheese layer. Add more cheese on top of the chicken to help hold the filling together as it melts.
Fold each tortilla in half to create a half-moon shape. Press gently to keep the filling in place.
Place the quesadillas onto the hot griddle. Cook for several minutes on each side until the tortillas become golden brown and crispy while the cheese melts completely inside.
Carefully flip using a spatula and continue cooking until both sides are evenly toasted and the filling is hot.
Transfer the cooked quesadillas to a cutting board and let them rest briefly before slicing.
Cut each quesadilla into wedges and drizzle lightly with additional hot honey if desired. Garnish with chopped chives and serve immediately.