Pat the chicken breasts dry with a paper towel. Slice each breast in half horizontally to create two thin slices. Sprinkle evenly with the salt and pepper.

Place the ½ cup of flour onto a large plate. Dredge each piece of chicken into the flour to coat. (You can use additional flour if necessary.)

In a large frying pan, heat one tablespoon of the olive oil. Add half the chicken in a single, uncrowded layer and sauté until browned a bit on both sides but not completely cooked through. Remove chicken to a clean plate. Repeat with another tablespoon of the oil and the remaining chicken.

Heat the final tablespoon of olive oil in the skillet. Add the shallot and garlic and cook, stirring frequently, until softened.

Add capers and cook a minute or two more.

Pour in chicken broth. Stir to incorporate and scrape up any browned bits from the bottom of the pan. Bring to a brisk simmer.

Meanwhile, use a fork to stir together the butter and remaining three tablespoons flour in a small bowl until a smooth paste forms. This is your beurre manié.

Add beurre manié to pan and use fork to whisk into sauce. Simmer for five minutes, until beginning to thicken.

Return chicken to pan, nestling into sauce. Continue simmering until chicken is just cooked through and sauce is silky and thickened.

Off the heat, stir in the lemon juice and sprinkle with the parsley. Garnish with lemon slices if you like.