PREHEAT OVEN to 400 degrees F and set rack at center. Line a quarter-sheet pan (9”x13”) with foil. Spray with cooking spray.
DRIZZLE THE CHICKEN BREASTS with olive oil and sprinkle generously with salt and pepper. Place at center of lined pan, leaving a little space between them.
STIR TOGETHER THE POTATOES, BROCCOLI, ONION, AND GARLIC in a large bowl with two tablespoons of olive oil, ¾ teaspoon kosher salt, ½ teaspoon fresh ground pepper, and a good pinch of crushed red pepper. Remove the broccoli to the cutting board. (You’ll add it later.) Spoon the remaining vegetables evenly around the chicken breasts.
ROAST ABOUT 45 MINUTES, remove from oven, and add broccoli on top of or between the vegetables. Pour the salsa on top of chicken breasts and sprinkle the vegetables (and chicken, if you like) with the Cheddar cheese. Return pan to oven and roast another 15 minutes or until an instant read thermometer placed into the center of the breasts registers 165 degrees F and vegetables are tender or tender-crisp for the broccoli. (Check small breasts at 10 min.)
REMOVE FROM OVEN and let the chicken and vegetables rest 5-10 minutes before serving hot. Store cooled leftovers well-wrapped in the fridge for up to 3 days. Do not freeze.