Mince the garlic and slice the lemon crosswise about 1/4-inch thick. Set aside.

Lay the chicken on a baking sheet and season with the salt and pepper. Put the flour in a pie pan and lightly dredge chicken on both sides, patting it gently to create the thinnest possible coating of flour.

In a large frying pan, heat the oil over medium heat. Add a single layer of chicken and cook until golden brown on underside, 3-4 minutes. Using tongs, flip the chicken and cook until lightly browned on the other side and cooked through, 3-4 minutes. Transfer to a clean baking sheet and repeat with remaining chicken, adding more oil if needed.

Using a wad of paper towels and tongs, carefully wipe any darker brown flour from the frying pan (but don't remove all the flour; it helps thicken the sauce). Add the lemon slices to the hot pan and cook over medium heat until golden brown on each side, turning once, 3-4 minutes. Add the garlic and cook, stirring occasionally, until softened, about 1 minute. Add the capers, wine, and chicken broth and bring the liquid to a boil, scraping up browned bits from bottom of pan.

Boil the sauce vigorously until reduced by half and thickened, about 5 minutes.

Return chicken to the pan and gently warm through, turning to coat in the sauce.

Transfer the chicken to a serving platter. Turn off heat under the sauce and add the butter. Let it melt gently, swirling the pan to combine.

Pour sauce over chicken, and top with parsley if you like. Serve immediately.