Cut each chicken breast in half lengthwise and place each piece of chicken into a large plastic zip-top bag or between two pieces of parchment paper. Use a mallet to gently pound them so they are even and about ½” (12.7mm) thick. Be careful that the meat does not break.

Add the flour to a shallow bowl. Season the chicken with salt and pepper, then dredge the chicken in the flour, shaking off any excess.

Heat a large skillet over medium heat, add the olive oil and heat through. Working in batches, add the chicken to the pan and fry until golden brown, about 2 minutes. Flip and continue to cook until cooked through, about an additional 2 to 3 minutes. Remove the chicken to a plate and repeat with remaining chicken.

Return pan to medium heat, add the shallots and sauté until tender, about 5 minutes. Add the garlic and sauté until fragrant, about an additional 30 seconds.

Add the wine and reduce by about half, about 5 minutes. Add the capers and lemon juice, then add the butter, 1 tablespoon (14g) at a time, swirling the pan to help emulsify the sauce, continuing to add the butter. The sauce should coat the back of a spoon. Stir in half the parsley.

Return the chicken to the pan, turning each piece to coat it in the sauce. Garnish with the remaining parsley and serve immediately.