While the sauce cooks, char the poblanos on a hot grill or over a gas stove, until blackened on all sides. Place the charred poblanos in a plastic bag or airtight container to steam for 10 minutes.
After the steam has loosened the charred skin, use a paper towel and your hands to remove the charred skin from the poblanos. At this point, you can also remove the seeds and stems, slicing the peppers open so they lay flat.
Pound the chicken breasts until evenly flattened to about ½-inch thickness.
Whisk together the all-purpose flour, garlic powder, kosher salt, and black pepper. In a large skillet, add enough olive oil to completely coat the bottom. Heat over medium high heat. Turn the oven down to 350°F.
Dredge each chicken breast lightly through the flour mixture, tapping off any excess, before carefully laying them into the skillet.Sear the chicken 3-4 minutes per side, until fully cooked, working in batches if necessary to avoid overcrowding the pan. Arrange the cooked chicken in a baking pan. Top each chicken breast with a piece of poblano pepper, then evenly sprinkle the cheese over the chicken breasts. Top with the finished tomato sauce.
Bake the chicken for 5-10 minutes, until the cheese is melted.
Serve with the spiced black beans, and enjoy.