Preheat the broiler. Put the poblanos cut-side down on a foil-lined broiler pan. Add the tomatoes, onion and garlic and broil until charred, 7 to 8 minutes. Cool slightly, then peel the garlic. Chop the garlic, onion and tomatoes; toss in a bowl with salt to taste. Peel the poblanos and cut each in half.
Preheat the oven to 350 degrees F. Brush the bun tops with the egg white. Mix the oregano, cumin and chili powder, sprinkle on the bun tops and put the tops and bottoms on a baking sheet. Bake 5 minutes.
Heat a large cast-iron skillet over medium-high heat. Shape the beef into four 3/4-inch-thick patties; make an indentation in the centers. Sprinkle the skillet with salt, then cook the burgers until browned on the bottom, about 4 minutes. Flip, then top each burger with some of the tomato mixture and cheese. Cover and cook 3 to 5 minutes. Serve on the spiced buns with the poblanos and the remaining tomato mixture.