Remove chicken breasts from refrigeration 20-30 minutes before cooking. Pat thoroughly dry with paper towels. Combine 1 tablespoon salt, pepper, garlic powder, and cumin, then season chicken breasts evenly on both sides.
Position oven rack in middle position and preheat to 375°F. Heat oven-safe skillet over medium-high heat until approximately 400°F surface temperature.
Add oil to the hot skillet. When oil shimmers, place chicken breasts presentation-side down. Sear undisturbed for 3-4 minutes until deep golden brown crust forms.
Flip chicken breasts, then immediately transfer skillet to preheated oven. Bake until internal temperature reaches 163°F at thickest point, approximately 8-12 minutes.
Remove skillet from oven. Transfer chicken to a clean plate and tent loosely with foil. Rest for 5-7 minutes.
While chicken rests, prepare mango salsa. Dice mangoes into ¼-inch cubes. Combine with red onion, bell pepper, jalapeños, cilantro, lime juice, olive oil, and remaining salt and pepper in a non-reactive bowl.
Place rested chicken breasts on serving plates. Top generously with prepared mango salsa. Serve immediately.