Preheat oven to 400°F. Heat 2 tablespoons of the oil in a large skillet over high. Sprinkle chicken thighs with 1/2 teaspoon each of the salt and pepper. Place 4 of the chicken thighs, skin side down, in hot skillet; cook until skin is golden and crispy, about 5 minutes. Turn, and continue cooking 1 minute. Transfer to a plate (chicken will not be cooked through). Repeat with remaining 2 tablespoons oil and 4 chicken thighs. Add onion and mushrooms to skillet, and cook over medium, stirring occasionally, until softened, about 4 minutes. Add anchovies and garlic; cook, stirring constantly, until fragrant, about 1 minute. Add wine, and cook until almost completely reduced, about 2 minutes. Stir in crushed tomatoes, sugar, and remaining 1/2 teaspoon each salt and pepper. Add basil sprigs, and bring to a boil. Remove from heat, and stir in vinegar and 3 tablespoons of the capers. Spoon half of the tomato mixture in a 13- x 9-inch baking dish. Nestle chicken thighs into mixture, skin-side up, and spoon remaining tomato mixture on top of thighs. Sprinkle with remaining 1 tablespoon capers. Bake until cooked through, and a thermometer inserted in thickest portion of chicken registers 160°F, 30 to 35 minutes. Remove from oven, and sprinkle with Parmesan and basil. Serve immediately, or store in refrigerator up to 3 days.