Heat 1 tablespoon oil in a large skillet over medium heat. Sprinkle chicken with ¼ teaspoon each salt and pepper. Cook the chicken, turning once, until browned and an instant-read thermometer inserted in the thickest part registers 165°F, about 6 minutes. Transfer to a plate and cover to keep warm.
Add the remaining 1 tablespoon oil to the pan. Add onion and the remaining ¼ teaspoon each salt and pepper; cook, stirring, until soft and starting to brown, about 4 minutes. Add sherry and cook, stirring and scraping up any browned bits, until most of the liquid has evaporated, about 1 minute. Whisk sour cream and broth together in a glass measuring cup. Pour into the pan along with any accumulated chicken juices; cook, stirring, until bubbling, 1 to 2 minutes. Return the chicken to the pan and turn to coat. Sprinkle with basil, if desired.
