Season the chicken: Pat chicken dry and sprinkle with salt and pepper. Dry skin helps it brown nicely.
Brown the chicken: Heat olive oil in a large skillet or Dutch oven over medium-high heat.
Place chicken skin-side down and sear until deeply golden, about 5–7 minutes per side. Transfer to a plate.
Cook the veggies: Reduce heat to medium. Add onion, peppers, and mushrooms to the pan with a pinch of salt.
Cook, stirring occasionally, until softened and lightly browned, about 6–8 minutes.
Aromatics and spices: Add garlic, red pepper flakes, oregano, thyme, and the bay leaf. Stir for 30 seconds until fragrant.
Deglaze with wine: Pour in the wine and scrape up the browned bits from the bottom. Let it simmer for 2–3 minutes to reduce slightly.
Add tomatoes and broth: Stir in crushed tomatoes and chicken broth.
Bring to a gentle simmer.
Return the chicken: Nestle the chicken pieces into the sauce, skin-side up. Spoon some sauce over, but keep the skin mostly above the liquid so it stays a bit crisp.
Simmer until tender: Cover and cook on low for 25–30 minutes, or until the chicken is cooked through and very tender. Uncover for the last 5–10 minutes to thicken the sauce.
Finish the flavor: Stir in capers and olives.
Taste and adjust salt and pepper. Remove the bay leaf.
Serve: Sprinkle with parsley. Serve over pasta or creamy polenta, or with warm bread to mop up the sauce.