Chicken cutlets cook in a matter of minutes, and dredging them in flour helps form a luscious pan gravy.
Season chicken with salt and pepper, then dredge in flour. Heat a large skillet over medium-high; swirl in 1 tablespoon oil. Cook chicken, flipping once, until golden brown in places and just cooked through, 4 to 5 minutes. Add capers, brine, and broth; bring to a boil, turning chicken a few times. Transfer chicken to a plate; cover to keep warm. Continue to boil sauce until reduced to 1/2 cup, 3 to 4 minutes. Transfer to a bowl; cover.
Return skillet to medium-high; swirl in remaining 2 tablespoons oil. Add squashes, season with salt and pepper, and cook, stirring occasionally, until golden brown in places, 3 to 4 minutes. Add spinach, a few handfuls at a time, and cook, stirring, until just wilted. Stir in feta; season with salt and pepper. Serve, alongside chicken topped with caper sauce, and bread and butter.