Preheat the broiler. Line a broiler pan with foil.
Put the poblanos cut-side down on the prepared pan and broil until charred, 7 to 8 minutes. Cool slightly, then peel the poblanos. Set aside.
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 center of each with your thumb. 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 Mexican Tomato Sauce, the Monterey Jack and a poblano half. Cover and cook until the burger is cooked and the Monterey Jack is melted, 3 to 5 minutes. Serve on buns with more Mexican Tomato Sauce on the side for dipping.
Combine the garlic, cilantro, tomatoes, serrano chile, cinnamon, onion and 1 tablespoon salt in a blender and puree until smooth.
Transfer the tomato puree to a medium saucepan and bring to a boil. Lower the heat and simmer until slightly thick, about 10 minutes.
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