Preheat oven to 325 degrees F (165 degrees C).

Sprinkle chicken with 1 teaspoon salt. Heat oil in a wide, heavy, oven-safe pot over medium-high heat until hot but not smoking, about 1 minute. Add chicken, skin sides down, and cook, turning once, until browned, about 10 minutes. Transfer to a plate.

Pour off all but 2 tablespoons fat. Reduce heat to medium and cook pepper, onion, and garlic, stirring, until softened, about 5 minutes. Stir in oregano, thyme, tomato puree, water, and remaining 3/4 teaspoon salt; bring to a simmer. Return chicken to pot, return to a simmer, cover, and transfer to oven. Bake until chicken is tender, 45 to 50 minutes.

Transfer chicken to a bowl and chill, uncovered, until cool enough to handle (20 minutes). Discard skin and bones. Pull meat into coarse shreds.

Add chicken, beans, and vinegar to tomato mixture in pot and simmer until heated through, about 5 minutes.

Meanwhile, cook pasta in salted water according to package directions. Drain, reserving 1 cup pasta water. Stir pasta into chicken mixture, thinning with pasta water as needed. Season with additional salt, if desired.