Preheat the grill.

In a 9-inch square baking pan, combine vegetable oil, lime juice, chopped green chilies, and chopped garlic to make the marinade.

Add the chicken breasts to the marinade, ensuring they are fully coated.

Marinate in the refrigerator for at least 45 minutes, turning once halfway through.

Remove the chicken from the marinade and let any excess drip off.

Grill the chicken over hot coals for about 7 minutes on each side, or until it is almost cooked through.

Once the chicken is nearly done, top each piece with 2 slices of Cheddar cheese and continue grilling until the cheese starts to melt.

Remove the chicken from the grill and serve immediately with salsa on the side.