Combine lukewarm water and salt in a large bowl, stirring until the salt dissolves. Add chicken cutlets and let stand at room temperature for 15 minutes. Remove chicken and pat dry, discarding the brine.

Season each cutlet with salt and pepper, then dredge in flour; shake off any excess flour. Place seasoned and floured cutlets on a plate.

Preheat the oven to 400°F (200°C) and grease a 9 x 13-inch baking dish.

Heat half the butter (4 tablespoons) and olive oil in a skillet over medium-high heat. Add 4 floured cutlets and cook until browned, about 3 minutes per side. Transfer to the baking dish. Repeat with the remaining cutlets.

Bake chicken cutlets in the oven until fully cooked, about 10 minutes. Ensure an internal temperature of at least 165°F (74°C).

For the sauce, heat the skillet over medium-high heat, adding chicken stock, lemon juice, and capers. Stir to scrape up browned bits. Incorporate minced garlic, salt, and pepper. Add remaining 4 tablespoons of butter one at a time, stirring constantly until the sauce reaches a spoonable consistency, about 4 minutes.