Heat some oil in a medium skillet, and add in the green peppers. Sauté until softened. Mix in the chicken and black beans. Pour in the beaten eggs. Scramble everything together until cooked, about 2 minutes.
Remove the pan from heat and stir in the shredded cheese.
Heat a large skillet on medium-high heat. Place on a tortilla. Evenly pour on the scrambled egg filling. Cover with a second tortilla and gently press down with a spatula. Cook for about 3 minutes, until the tortilla is golden brown on the bottom.
Quickly check the inside of the quesadilla. The cheese should have melted enough that it is holding the ingredients together. Quickly and carefully, flip the quesadilla over. If some of the ingredients fall out, place it back into the quesadilla. Cook until the other side is golden brown, about 3 minutes.
Transfer to a cutting board and cut into wedges to serve. Top with your favorite toppings and a squeeze of lime.
You may also add a bit of oil to the bottom of the skillet when you cook the quesadilla. I prefer not to because I don't like the quesadilla greasy/oily.