Heat the oil in a large skillet over medium-high heat. Add the onion, bell peppers, ½ teaspoon salt, and ¼ teaspoon pepper. Cook, stirring occasionally, until softened and starting to brown, about 8 minutes. Remove from the pan and set aside. (Alternatively, you can do this step on a cast iron skillet on the grill.)
Mash the avocados with the salsa and lime juice. Season to taste with salt and pepper.
Prepare the grill for direct heat and preheat to medium-high.
Combine the chicken and cheese in a large mixing bowl. Divide into 4 equal portions. Form each portion into patties and make a deep impression in the center of each patty with your thumb. Brush the patties with oil and season with salt and pepper.
Grill the burgers for 4 minutes with the lid closed. Flip and cook another 4 minutes or until the burgers are cooked through.
Serve the burgers on buns spread with the avocado mixture and topped with the onion and pepper mixture.