Heat olive oil in a large skillet over medium-high heat. Season chicken breasts with salt and pepper. Place chicken in the skillet and cook until browned, about 5 minutes per side.
Reduce heat to medium; add butter and garlic. Cook and stir mixture until garlic is fragrant, about 3 minutes. Mix olives, gin, vermouth, and lemon juice into chicken mixture; simmer until chicken is no longer pink in the center and sauce is thickened, about 5 minutes. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C).