Preheat the grill to medium-high. Cut the beef into 1-inch cubes, then place on a plate, wrap in plastic wrap, and chill in the refrigerator for about 30 minutes, until cold.
Meanwhile, put the onion, lemon juice, parsley, mint, garlic, and crushed red pepper, if using, into a small bowl, season with salt and pepper, and combine. Stir in the oil and set aside.
Place half the beef in a food processor or blender and pulse about 15 times. Season with salt and pepper and pulse an additional 10-15 times until the meat is finely chopped but not overprocessed. Remove from the processor and repeat with the remaining meat.
Divide the meat into four equal portions and shape each portion into a patty.
Place the patties on the grill and cook for 3 minutes on each side for medium-rare, and 4 minutes on each side for medium.
Place the burgers in the rolls. Top with avocado slices and a few spoonfuls of the sauce. Serve immediately.