Preheat a large nonstick skillet to medium-high heat. Chop the chicken into bite-size pieces and set aside. When the skillet is hot, add the sliced onions and sear for 2 minutes to soften and char the edges, flipping once. Move the onions to a plate and sear the sliced peppers for 2 minutes until slightly charred. Move the peppers to the plate as well.
Add the butter to the skillet. Once melted, add the chicken pieces. Salt and pepper liberally, then brown for 2 minutes per side. Add the apple butter, cayenne pepper sauce, and vinegar. Stir to combine and let the mixture simmer another 2 minutes. Then turn off the heat and stir to coat the chicken in the sauce. Move the chicken to the holding plate as well.
Lower the heat to medium-low and add the first tortilla to a clean nonstick skillet. Sprinkle it with a heaping 1/4 cup cheese. Spread one-quarter of the onions, peppers, and chicken over the top. Then sprinkle with fresh cilantro. Sprinkle another 1/4 cup cheese over the top and cover with a second tortilla.
Wait to flip the quesadilla until the bottom tortilla is golden brown, and the cheese has started to melt to hold everything together. Place a flat spatula under the bottom tortilla, and another on the top tortilla. Then quickly flip the quesadilla. Brown the bottom tortilla, then move to a warming drawer or warm oven to keep crisp. Repeat with the remaining ingredients. Once all 4 quesadillas of cooked, cut them into wedges and serve warm!