Season chicken all over with garlic salt and black pepper.

Melt 1 tablespoon butter and olive oil in a large skillet over medium heat; cook chicken in hot butter until browned, about 4 minutes per side. Transfer chicken to a plate.

Melt remaining butter in the same skillet; cook and stir mushrooms, garlic, salt, and black pepper together in hot butter until mushrooms are tender, 5 to 10 minutes. Pour cider into the skillet and bring to a boil while scraping the browned bits of food off of the bottom of the pan with a wooden spoon. Cook until cider reduces, 3 to 5 minutes.

Return chicken to skillet, reduce heat to medium, cover the skillet with a lid, and cook until chicken is no longer pink in the center and the juices run clear, 20 to 25 minutes. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C). Place 1 slice of Swiss cheese on each piece of chicken, cover the skillet with a lid, and cook until cheese melts, 1 to 2 minutes.