Lay chiles on grill racks and turn flame on high. Char the chiles, turning them with tongs, until skins are blackened, 3 to 8 minutes. Transfer chiles to a bowl and let stand, covered, until cool enough to handle. Peel the chiles by rubbing off the skin. (If you have sensitive skin, it helps to wear plastic gloves when handling chile peppers.) Cut off tops and discard seeds and ribs, and dice the chiles.
Heat oil in a large skillet over medium heat. Add onions and cook for 6 to 8 minutes until just beginning to brown. Add the diced, roasted poblanos and garlic. Cook, stirring, for 5 minutes. Add the oregano and 1/4 teaspoon salt and cook for another minute. Remove from heat.
Divide the meat mixture into 4 equal portions, gently shaping each into a 1/2-inch-thick patty. Press a nickel-sized indentation in the center of each patty. Sprinkle the remaining 1/4 teaspoon salt and black pepper evenly over the burgers.
Place patties on a grill rack and grill 4 minutes on each side or until desired degree of doneness. Top each patty with 3 tablespoons of the warm poblano mixture and 2 tablespoons cheese during last 2 minutes of cooking. Cover and cook until the patty has desired doneness and cheese is melted.
In a small bowl, whisk together the mayo, chili powder and lime juice.
Place buns, cut sides down, on grill rack; grill for 1 minute. Spread the cut sides of the rolls with the mayo mixture. Place bottom halves of buns on 4 plates; top each with patty, lettuce, tomato, onion, top of bun.