Mince 2 garlic cloves. Juice 1 large lemon until you have 3 tablespoons. Pick the leaves from 8 to 10 fresh parsley sprigs and finely chop until you have 2 tablespoons. Place 1/4 cup all-purpose flour in a shallow bowl or large plate.

Cut 2 large boneless, skinless chicken breasts in half horizontally (also known as butterflying) to get 4 pieces total. Pat dry with paper towels. Season all over with 1 teaspoon of the kosher salt and 1/2 teaspoon black pepper. Working with one piece at a time, dredge the chicken in flour and shake off any excess. Place on a plate.

Heat 2 tablespoons of the olive oil and 1 tablespoon of the unsalted butter in a large skillet over medium heat until shimmering. Add half of the chicken in a single layer and sear until browned, 3 to 5 minutes per side. Transfer to a large plate (the chicken will not be cooked through).

Add the remaining 2 tablespoons olive oil and 1 tablespoon of the unsalted butter to the pan and repeat browning the remaining chicken. Transfer to the same plate.

Reduce the heat to medium. Add the garlic and cook until fragrant, 20 to 30 seconds. Add 1/4 cup dry white wine and cook until reduced by about half, scraping up any browned bits from the bottom of the pan with a wooden spoon. Add 3/4 cup low-sodium chicken broth and bring to a simmer.

Reduce the heat to maintain a gentle simmer. Add the lemon juice, remaining 1/4 teaspoon kosher salt, remaining 2 tablespoons unsalted butter, and 3 tablespoons drained nonpareil capers, and stir to combine. Once the butter is melted, return the chicken and any accumulated juice on the plate to the pan. Simmer until the chicken is cooked through and registers at least 165ºF on an instant-read thermometer, 4 to 6 minutes. Taste and season with more kosher salt as needed. Garnish with the parsley.