If using whole chicken breasts, place them on a small baking sheet and freeze for 15 minutes for easier slicing.

Carefully slice each breast horizontally into cutlets.

In a large zip-lock bag or bowl, dissolve kosher salt in 1 quart of lukewarm water. Seal and shake to mix well, then add the chicken cutlets. Brine at room temperature for 15 to 30 minutes while you prep other ingredients. Remove the chicken, rinse, drain, and pat dry with paper towels. Season with ground black pepper.

Place flour on a large plate. In a large non-stick skillet, heat 1 tbsp butter and 1 tbsp olive oil over medium-high heat until melted. Dredge the chicken cutlets in the flour, shaking off the excess, and add half to the skillet. Cook without overcrowding, until browned on one side, about 5 minutes. Flip and cook for another 3 minutes on the other side. Transfer to a plate and cover with foil to keep warm. Repeat with remaining butter, olive oil, and chicken.

In the now-empty skillet, add garlic and shallots, sautéing until fragrant, about 1 minute. Pour in the white wine, deglazing the pan by scraping up any browned bits. Add capers, chicken broth, and lemon zest strips. Cook over medium-high heat, stirring often, until the sauce reduces by half and starts to thicken. Discard the lemon zest strips. Return the chicken and any juices to the pan, heating through for a few minutes.

Remove the skillet from the heat, stir in the remaining 2 tbsp butter and lemon juice until the butter melts. Serve over cooked spaghetti, garnished with additional lemon slices and parsley if desired.