Preheat the oven to 350 degrees . Using a knife, split each chicken breast horizontally, being careful not to cut all the way through, then lay flat. Place between 2 sheets of plastic wrap. Using a meat mallet, pound the breasts 1/2 inch thick.
Season the chicken with salt and pepper. Place a slice of provolone on each breast, then top with 3 apricot halves. Fold to enclose and cover with a sheet of plastic wrap; pound the edges together to seal. Coat the breasts with flour.
In a nonstick skillet, heat the olive oil over high heat. Add the chicken breasts and rosemary and cook, turning once, until golden, 2 to 3 minutes. Transfer to a rimmed baking sheet; discard the rosemary.
Add the white wine and lemon juice to the same skillet and cook, scraping up any bits, over medium-high heat for 2 minutes. Stir in the butter until melted. Pour the sauce over the chicken, cover with foil and bake for 10 minutes; uncover and bake until the sauce is thickened, about 10 minutes more. Serve with the rice; sprinkle with pepper.