Preheat the oven to 350°F. Slice the chicken breast horizontally, as if to butterfly each piece, but not cutting all the way through. Lay each flat on a work surface. Cover with a piece of plastic wrap and use a meat mallet to pound the breasts 1/2 inch thick.
Season the chicken with salt and pepper, to taste. Place a slice of cheese on each breast, then top with 3 or 4 apricot halves. Close up the chicken pieces and cover again with a sheet of plastic. Lightly pound the edges together on each piece as to seal together.
Spread out the flour on a plate. One at a time, place the chicken on top of the flour and gently pat; flip once to coat the other side.
Heat the olive oil over high heat in a nonstick skillet. Add the chicken breasts and cook, turning once, until golden, about 2 to 3 minutes a side. Transfer to a rimmed baking sheet, leaving cooking juices in the skillet.
To the same skillet, add the wine and lemon juice; cook, scraping up any bits, over medium-high heat for 2 minutes. Add the butter and stir until melted.
Pour the sauce over the chicken, cover with foil and bake for 10 minutes; uncover and bake another 10 minutes or until cooked through.
Serve immediately and any remaining pan juices drizzled over the top.