Slice the chicken breasts in half horizontally to create 4 thin cutlets. Place them between two sheets of plastic wrap and gently pound them to an even 1/3-inch thickness. Season both sides with 1/2 teaspoon salt and black pepper.

In a shallow dish, whisk together the flour, garlic powder, and a pinch of salt. Lightly dredge each chicken cutlet in the flour mixture, shaking off any excess. Dredging creates that signature golden exterior crust that grabs onto the sauce.

Heat 2 tablespoons of olive oil and 1 tablespoon of butter in a large 12-inch heavy stainless steel or cast-iron skillet over medium-high heat. Once hot and shimmering, add the cutlets in a single layer and sear undisturbed for 3 to 4 minutes per side until deeply golden brown and cooked through to 165°F (74°C). Transfer cutlets to a warm plate.

Add the lemon slices to the hot skillet and sear for 1 minute until caramelized on the edges; remove to the plate with the chicken. Lower the heat to medium, add the minced garlic, and sauté for 30 seconds until fragrant.

Pour in the chicken bone broth, scraping up all the golden browned bits (fond) stuck to the bottom of the skillet with a wooden spoon. Bring to a simmer and let reduce by half, about 3 minutes.

Stir in the heavy whipping cream, fresh lemon juice, lemon zest, drained capers, and fresh thyme. Simmer gently on medium-low for 2 minutes until the sauce slightly thickens into a glossy, velvety consistency.

Remove skillet from direct heat and swirl in the remaining 2 tablespoons of cold cubed butter until fully emulsified and luxurious. Return the seared chicken cutlets and caramelized lemon slices to the skillet, spooning the warm sauce over them. Garnish with fresh chopped parsley and serve immediately!