Rinse the onion in a fine mesh strainer under cold water and let the excess water drain off. In a medium bowl mash the avocado and mayonnaise together with a fork. Stir in the onion, tomatoes, cilantro, lime juice, and garlic. Season generously with salt. Cover with plastic wrap, pressing the wrap directly onto the surface, and set aside. (The mayonnaise can be prepared up to 8 hours ahead.)
Prepare the grill for direct cooking over high heat (450° to 550°F).
Brush the cooking grates clean. Grill the chiles over direct high heat, with the lid closed as much as possible, until the skin is blackened on all sides, about 10 minutes, turning occasionally. Transfer the chiles to a bowl and cover with plastic wrap to trap the skin. Let stand for about 10 minutes. Remove the chiles from the bowl and peel off and discard the blackened skin, stems, and seeds. Chop the chiles into ½-inch dice.
In a large bowl gently mix the ground chuck, chiles, salt, and pepper, and shape into four patties of equal size and thickness, about ¾ inch thick. With your thumb or the back of a spoon, make a shallow indentation about 1 inch wide in the center of each patty so the centers are about ½ inch thick. This will help the patties cook evenly and prevent them from puffing on the grill.
Grill the patties over direct high heat, with the lid closed as much as possible, until cooked to medium doneness, 8 to 10 minutes, turning once when the patties release easily from the grate without sticking. During the last minute of cooking time, toast the buns, cut side down, over direct heat. Top the burgers with the mayonnaise and serve warm.
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