Pound chicken breast to 1/2" thickness.
Mix olive oil, lime juice, and chili powder. Place chicken breast(s) in, flip to get full coverage, and marinate for 15-60 minutes.
Heat a pan on medium heat. Deploy chicken breast. Cook for 3-4 minutes per side. Shortly before the second side is done, place cheddar on and cover pan to melt it.
Put on a toasted or warmed bun/roll/bagel. Add salsa.