Pat the chicken dry with paper towels and season with salt and pepper. Heat the oil in a large Dutch oven over medium-high heat until just smoking. Add the chicken skin-side down and cook until golden, about 5 minutes. Transfer the chicken to a plate, leaving the fat in the pot.
Add the onion, pepper flakes, and 1/2 teaspoon salt to the fat in the pot and place over medium heat. Cook, scraping up any browned bits on the pan bottom, until the onion is softened, about 5 minutes.
Stir in the garlic and cook until fragrant, about 15 seconds. Stir in the rice and cook until the edges turn translucent, about 3 minutes. Stir in the broth and wine and bring to a simmer.
Return the chicken to the pot, skin-side up. Cover, reduce heat to low, and cook until the thickest part of the breast registers 160 degrees on an instant-read thermometer, about 25 minutes.
Transfer the chicken to a serving platter. Stir the black beans, corn, cheddar, and cilantro into the rice. Let the cheddar melt for 1 minute. Season the rice mixture with salt and pepper to taste. Serve with the salsa.