Season the chicken breasts with salt and pepper.
Heat the olive oil in a large skillet over medium-high heat and add the chicken.
Cook until golden brown, about 5-7 minutes, then flip and cook the opposite side until golden brown and cooked through, about 7 more minutes or until the internal temperature reaches 165°F on a meat thermometer inserted into the center of the thickest portion. Remove the chicken from the skillet and place on a plate. Tent with foil to keep warm.
Return the skillet to medium heat. Add a drizzle of oil, if necessary, and the diced shallots.
Cook, stirring frequently and scraping up and browned bits from the chicken, until softened, about 2-3 minutes.
Stir in the olives and cook for a minute or two, until warm.
Mix in the lemon juice, lemon zest, and basil.
Taste. If the sauce is too sour for your taste, you can add water, 1 tablespoon at a time, until it’s more to your liking.
Serve the chicken topped with a spoonful or two of the lemon olive sauce.