In a small bowl, combine the onion with 2 tablespoon vinegar, season with salt and pepper and set aside.
Heat 2 tablespoon olive oil in a large skillet over medium-high. Add bread, season with salt and pepper. Cook, tossing, until golden brown, 4-6 minutes. WATCH CAREFULLY. Transfer to a medium bowl. Wipe out skillet.
Between two pieces of plastic wrap, using the flat side of a meat mallet, pound each chicken thigh until about ¼" thick; season with salt and pepper. If using bone-in chicken breasts, with a thin, sharp knife, cut bones and cartilage from out. Leaving the skin on either the thighs or breasts gives a satisfying layer of fatty crunch.
Heat 1 Tbsp. olive oil in skillet over medium-high. Cook 2 chicken thighs or breast, at a time (if using chicken with skin, skin side down), until golden brown and nearly cooked through, about 4 minutes. Turn and cook until cooked through, about 1 minute more; second side will not brown. Transfer to a platter. Repeat with remaining cutlet and 1 Tbsp. olive oil (no need to wipe out skillet).
Cut half of tomatoes in half. Heat 2 Tbsp. olive oil in same skillet over medium-high. Add whole tomatoes, season with salt and pepper. Cook, tossing occasionally, until lightly blistered and starting to burst, about 5 minutes. Toss in sugar (or agave) and remaining 1 Tbsp. vinegar. Transfer to bowl with croutons. Add pickled onion with pickling liquid, halved tomatoes, parsley, and remaining 2 Tbsp. olive oil and toss. Serve chicken with panzanella spooned over.