Place a sheet of plastic wrap over an ice cube tray. Scoop 1 tbsp. cheese dip into 4 of the wells. Cover and freeze until firm, about 1 hour. Lift the plastic from the tray. Flatten each cube into a 1 1/2-inch disk and return to freezer.
Heat a griddle pan, a large, heavy skillet or a grill over medium-high. In a large bowl, gently mix the ground beef, bacon, chilies, kosher salt, cumin and a few grinds of pepper; form into 4 balls. Press your thumb into the center of each ball, creating a large indentation; press 1 cheese disk into each. Shape the beef mixture around the cheese disk; pinch to seal. Flatten to form 3/4-inch-thick patties. Cook right away, about 3 minutes per side for medium-rare. Squeeze lime wedges over the patties.
Divide the pepper jack among the bun bottoms. Add the patties, onion, avocado, cilantro and bun tops.
