Pat all the chicken pieces dry with paper towels and season generously on both sides with salt and pepper. Drying the chicken is important because moisture is the enemy of a good sear, and a good sear is where this dish starts to build its flavor.

Heat the olive oil in a large deep skillet over medium-high heat until shimmering. Brown the chicken in batches, skin side down first. Do not crowd the pan or the pieces will steam instead of brown. Cook each batch for 4 to 5 minutes per side until the skin is a deep golden brown. Transfer browned pieces to a plate and repeat with the remaining chicken.

Reduce the heat to medium. Add the sliced onion and both bell peppers to the same skillet. Cook for 4 to 5 minutes, stirring occasionally, until the vegetables have softened and picked up a little color. Add the minced garlic and tomato paste and stir for 1 minute until the tomato paste darkens slightly and the garlic is fragrant.

Pour in the crushed tomatoes and chicken broth. Stir well, scraping up any browned bits stuck to the bottom of the pan. Add the olives, oregano, and dried basil. Stir to combine and taste the sauce for salt, adjusting as needed.

Nestle all of the browned chicken pieces back into the skillet in a single layer, skin side up if possible. Spoon a little sauce over each piece. Bring the liquid to a gentle simmer, then reduce the heat to medium-low and cover the skillet with a lid.

Braise for 35 to 40 minutes, until the chicken is cooked through and reads 165 degrees F at the thickest part, and the sauce has thickened and deepened in color. Check once or twice and stir the sauce gently, making sure nothing is sticking to the bottom.

Remove the lid for the last 5 minutes of cooking to let the sauce reduce a bit more if it seems too thin. Scatter fresh parsley over the top before serving. Serve over egg noodles, creamy polenta, or with thick slices of Italian bread to soak up the sauce.