Sprinkle both sides of the burgers with fajita seasoning and cook according to the package directions. During the last minute of cooking, top each burger with a slice of cheese, cover the burgers, and cook until the cheese has melted.
While the burgers are cooking, heat the oil in a large skillet over medium-high heat and add the peppers and onion. Season with a large pinch of salt and pepper. Cook, stirring occasionally, until the vegetables are softened and beginning to brown, about 8 minutes.
Smash the avocados with lime juice and season to taste with salt and pepper.
Spread avocado on both sides of the hamburger buns, add the burger, and top with vegetables.