Combine all of hot green chile paste ingredients in food processor and mix to a coarse paste. Do not overprocess.
In a large mixing bowl, combine green chili paste and ground chuck. Carefully distribute paste into meat, but do not over handle. Cover with plastic wrap and refrigerate.
To prepare cool green avocado cilantro sauce, begin by chopping onion and garlic in food processor. Cut avocado in half, remove pit and scoop flesh into processor. Add salt, black pepper, mayonnaise, and lemon juice; combine until smooth. Add cilantro leaves and briefly pulse. Place finished sauce into plastic bowl with lid. Press plastic wrap directly onto sauce, cover with lid and refrigerate until ready to assemble the burgers.
Remove meat from refrigerator. Use food scale to divide meat into six equal portions. Form meat into patties to fit the size of the hamburger buns. Wrap each patty in an aluminum foil packet and refrigerate until grill is ready.
Preheat gas grill to medium high.
Prepare toppings by slicing the onion and tomato into 6 medium slices, one slice of each per burger. Wash and dry lettuce leaves. Select six large leaves and chill until burger assembly.
Split buns and lightly brush insides with olive oil.
Remove patties from refrigerator and unwrap. Lightly brush grill with oil to prevent sticking. Grill patties for 5 minutes on one side and 3-4 minutes on second side for medium doneness.
During last few minutes of grilling, place buns with oiled side down and lightly toast on the grill.
Assemble burgers by generously spreading avocado sauce on the toasted sides of each bun, then place lettuce leaf on bottom bun, followed by meat patty, onion slice, tomato slice and top bun. Serve burgers.