Pat the chicken pieces dry and season with 1/2 teaspoon of the salt and the pepper. Heat 1 tablespoon of the olive oil in a large deep skillet over medium-high heat until shimmering. Add the chicken in a single layer and sear, turning once, until golden on two sides, about 6 minutes. (It doesn't need to be cooked through yet.) Transfer to a plate.

Add the remaining 1 tablespoon olive oil to the skillet. Stir in the onion, both bell peppers, and the mushrooms; cook until softened and the mushrooms have given up their liquid, about 4 minutes. Add the garlic and cook 30 seconds more, until fragrant.

Stir in the tomato paste and cook 1 minute, until it darkens slightly and coats the vegetables. Pour in the red wine (or broth), scraping up every browned bit from the bottom of the pan, and simmer 2 minutes to cook off the raw alcohol edge.

Stir in the crushed tomatoes, Italian seasoning, red pepper flakes, and olives (if using), plus the remaining 1/2 teaspoon salt. Return the chicken and any accumulated juices to the pan. Bring to a lively simmer, then reduce to a steady bubble and cook, uncovered, 7–8 minutes, until the chicken is cooked through and the sauce has thickened slightly.

Remove from the heat and stir in the Parmesan until melted into the sauce. Taste and add another pinch of salt if needed — the olives and cheese both add saltiness, so go easy.

Spoon generously over hot polenta, pasta, or mashed potatoes. Shower with torn basil and extra Parmesan, and bring the skillet to the table.