garlic: peeled and wrapped in aluminum foil large red onion: sliced crosswise into 0.6cm ring
Gently mix lamb, 1 tsp. cumin, salt, and pepper in a bowl; divide mixture into 4 patties about 0.6cm thick. Transfer patties to a plate and cover with plastic wrap; refrigerate until ready to grill.
Heat a charcoal grill or set a gas grill to high; bank coals or turn off burner on one side. Grill garlic until tender, 8-10 minutes; remove from foil and transfer to a food processor. Mix onions and ¼ cup olive oil in a bowl; grill onions, flipping once, until charred in spots and tender, 8-10 minutes. Transfer onions to food processor along with yolks, lemon juice and zest, salt, and pepper; purée until smooth. With the motor running, drizzle in ½ cup olive oil, plus canola oil until emulsified. Transfer aïoli to a bowl; cover and refrigerate. Cut avocados in half lengthwise; discard pits. Scoop out flesh and mash with remaining cumin, plus lime juice, salt, and pepper in a bowl; cover and refrigerate.
Return grill to high heat. Brush buns with remaining olive oil; grill until lightly toasted, 1-2 minutes and transfer to a serving platter. Grill burgers on hottest part of grill, flipping once, until cooked to desired doneness, about 10 minutes for medium rare. If outside starts to burn before burgers are fully cooked, move to cooler section until done. Serve on buns with aïoli, avocado, lettuce, tomatoes, and top with fried eggs.