Place chicken in a large ziplock plastic bag; add salt, black pepper, and 1 ½ tablespoons pickled pepper brine. Seal bag, and massage until chicken is evenly coated with brine mixture. Refrigerate at least 1 hour or up to 8 hours. Remove chicken from brine; discard brine.

Heat oil in a large Dutch oven over medium-high. Add mushrooms; cook, stirring occasionally, until browned in spots, about 5 minutes. Reduce heat to medium. Add onion; cook, stirring often, until softened, about 3 minutes. Add bell pepper, garlic, and chopped cherry peppers; cook, stirring often, until fragrant, about 1 minute.

Add wine to mixture in Dutch oven. Bring to a simmer over medium-high; simmer, stirring occasionally and scraping up any browned bits from bottom of pot, until reduced by half, about 2 minutes. Add tomatoes and juices, oregano, and chicken; return to a simmer. Cook, stirring occasionally, until chicken is tender, tomato mixture thickens, and a spoon leaves a trail when dragged across bottom of pot, about 20 minutes.

Meanwhile, cook pasta according to package directions for al dente. Drain, reserving 1 cup cooking liquid. Set aside.

Stir heavy cream and remaining 1 ½ tablespoons pickled pepper brine into mixture in Dutch oven. Stir in cooked pasta; cook over low, stirring constantly, until tomato mixture clings to pasta, about 2 minutes. Season with additional salt to taste. If needed to thin sauce, stir in reserved pasta cooking liquid, ¼ cup at a time. Divide pasta evenly among shallow bowls. Garnish with cheese and black pepper.