Preheat your grill to medium-high heat.
In a medium bowl, mix together ground beef, onion, jalapeño, chili powder, cayenne pepper, salt, and black pepper.
Split the mixture in four and form four patties. Set aside.
In a small bowl, stir together avocado and lime juice. Season with salt and pepper to taste. Set aside.
Once the grill is hot, cook patties to desired temperature, about 2 to 3 minutes per side for medium rare. In the last minute of cooking, top each patty with a slice of pepper jack cheese and close lid until cheese melts.
Divide the mashed avocado between the bottom halves of the four buns. Place the cooked patties on top of the mashed avocado.
Top the burgers with salsa and fried onions. Finish with the top halves of the buns. Serve immediately.