Preheat the broiler to high.
Begin by making the green chile salsa: Place the chiles on a roasting tray with the garlic and onions. Drizzle with olive oil, season with salt, pepper and cumin and then place under the broiler to cook, 5 to 7 minutes. Once the skins have blistered and the onions have slightly charred around the edges, remove from the oven. Let cool slightly, then remove and discard the skins of the large chiles, along with the seeds and stems. Place the chile flesh, onions and garlic in a food processor. Add the lime juice and pulse 5 to 6 times to roughly chop the vegetables up; do not over chop as you want the salsa to be chunky and still have lots of texture where you can see pieces of the chile. Taste and season once more, then set aside as you prepare the burgers.
For the burgers: Preheat an outdoor grill and wipe down the grates with an oil-blotted paper towel. This not only cleans the grill but creates a nonstick surface.
Form the ground brisket into four 8-ounce patties about 1/2-inch thick. Make them slightly bigger in diameter than your buns, as they shrink when they cook. Also, add a dimple in the center as the burger puffs up in the middle when cooked, and this will ensure a flat, even patty. Sprinkle both sides with salt and pepper, and then place on the hot part of the grill. Cook for 5 to 7 minutes on one side, then flip the patties and top with the cheese. Close the grill or place a metal bowl over the top to melt the cheese and cook for 2 to 3 more minutes. Cut open the hamburger buns and lightly toast on the grill.
To assemble, place the lettuce, onion and tomato on the base of the burger bun. Top with a grilled patty and place a heaping spoonful of roasted green chile salsa on top. Smear the top burger bun with mayo, if desired, and place the top half of the burger bun on top. Serve.