Preheat oven to 350 degrees F. Spread some salsa in the bottom of a 9-inch x 13-inch baking dish. Set aside.
Season both sides of the chicken with salt and pepper.
Place a large skillet over medium-high heat. Add 1 tablespoon each olive oil and butter to the pan. Once the butter is melted, sear the chicken on each side for about 3-4 minutes. The chicken should be browned, but not cooked through. (It'll finish in the oven.)
Transfer the seared chicken to your prepared baking dish.
To the same skillet you cooked the chicken in, add 1 tablespoon each olive oil and butter. Once the butter is melted, add the sliced onions. Cook until the onions are softened and just starting to brown, about 5 minutes. Spread cooked onions over the chicken breasts.
In a mixing bowl, stir together the sour cream, 1/2 cup salsa, 3/4 cup cheddar cheese, and 3/4 cup jack cheese. Spread the sour cream mixture over the onions on the chicken.
Top each chicken breast with 2 tablespoons of salsa and spread to cover. Sprinkle with remaining cheddar and jack cheese. Top with any additional toppings - like jalapenos, olives, or green chiles.
Bake uncovered for 30 minutes.
Once the cheese is melted and the sauce is bubbling, remove the chicken from the oven. Sprinkle with chopped cilantro and serve warm.