With a boning knife, cut a wide pocket into the thickest part of each chicken breast half, taking care not to cut all the way through. Season the chicken evenly on both sides with 1/2 tsp. salt and 1/4 tsp. pepper total.
In a small bowl, mix the cheese, olives, and 1 tsp. each lemon zest and rosemary. Stuff the pockets with the cheese mixture and pin each shut with two toothpicks. Spread the flour in a shallow bowl and dredge the chicken in the flour, shaking off the excess.
Heat 1 Tbs. of the butter and the oil in a 12-inch heavy-duty skillet over medium-high heat until the foam from the butter subsides. Add the chicken and cook, flipping once and adding another 1 Tbs. of butter halfway through cooking, until golden-brown and cooked through (reduce the heat to medium if necessary), 14 to 16 minutes total. Transfer to plates and remove the toothpicks.
While the chicken cooks, melt 1 Tbs. of the butter in an 8-inch skillet over medium-high heat. Add the shallots and cook, stirring, until softened, about 3 minutes. Add the vinegar and boil until syrupy, about 3 minutes. Add the broth and boil for 1 minute more. Off the heat, whisk in the remaining 3 Tbs. butter, 1 tsp. zest, and 1/2 tsp. rosemary. Serve the chicken drizzled with the sauce.
Serve with garlicky sautéed spinach and crusty bread.
nutrition information (per serving):
31, Fat Calories
280, Saturated Fat
33, Monounsaturated Fat
10, Polyunsaturated Fat
Photo: Scott Phillips