Whisk together 2 tablespoons smoked paprika, 1 teaspoon garlic powder, ¾ teaspoon salt, and 3 tablespoons olive oil in a small bowl until a uniform paste forms.
Trim the chicken breasts of any fat or sinew. Lay them on a large sheet of plastic wrap with about 2 inches of space between each piece, then drape a second sheet of plastic wrap over the top.
Pound the breasts with a mallet until they reach an even thickness across the whole surface and are about 1½ times their starting size. This step ensures even cooking under the broiler.
Brush a baking sheet with 1 tablespoon of olive oil. Place the pounded breasts on the sheet and use a pastry brush to apply a thin, even coat of the paprika paste to both sides of each piece. Let the chicken rest at room temperature for 20 minutes so the seasoning can penetrate the meat. While waiting, position your oven rack at least 6 inches below the broiler element and preheat the broiler to high.
Broil the coated breasts for 6 to 7 minutes until the edges begin to firm up and crust. If the chicken appears to be cooking too fast or darkening too much, drop the rack one level lower. Flip each breast using tongs — if they look dry on top, dab on a bit more olive oil. Broil the second side for another 5 to 7 minutes until the internal temperature hits 160°F.
Rest the cooked breasts for 1 to 2 minutes before cutting into them, then slice and serve.