1To prepare roasted lemons: Preheat oven to 325°F. Line a baking sheet with parchment paper. Arrange lemon slices in a single layer on it. Brush the lemon slices with 1 tablespoon oil and sprinkle with ⅛ teaspoon salt. Roast the lemons until slightly dry and beginning to brown around the edges, 25 to 30 minutes.
2Meanwhile, prepare chicken: Cover chicken with plastic wrap and pound with a rolling pin or heavy skillet until flattened to about ½ inch thick. Sprinkle the chicken with ⅛ teaspoon salt and pepper. Place flour in a shallow dish and dredge the chicken to coat both sides; shake off excess (discard remaining flour).
3Heat 2 teaspoons oil in a large nonstick skillet over medium-high heat. Add the chicken and cook until golden brown, 2 to 3 minutes per side. Add broth and bring to a boil, scraping up any browned bits. Stir in capers. Boil until the liquid is reduced to syrup consistency, 5 to 8 minutes, turning the chicken halfway. Add the roasted lemons, butter, 2 tablespoons parsley and more pepper, if desired; simmer until the butter melts and the chicken is cooked through, about 2 minutes. Transfer to a platter. Sprinkle with the remaining 1 tablespoon parsley and serve.
Make Ahead Tip: Cover and refrigerate the roasted lemons (Step 1) for up to 2 days.