Heat 1 1/2 teaspoons oil in a large skillet over medium-high. Sprinkle chicken with 1/2 teaspoon salt and 1/2 teaspoon black pepper. Add chicken to pan; cook 3 minutes on each side or until browned. Place, skin side up, on one side of a rimmed baking sheet.
Combine remaining 1 1/2 teaspoons oil, remaining 1/4 teaspoon salt, remaining 1/4 teaspoon pepper, and broccoli in a bowl; add to other side of pan with chicken. Bake at 425°F for 10 minutes or until chicken is done.
Return skillet to medium-high. Add onion and bell peppers; sauté 3 minutes. Stir in garlic, thyme, and anchovy paste; cook 1 minute. Stir in stock, scraping pan to loosen browned bits. Stir in vinegar. Serve with chicken and broccoli.
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