Place chicken in a gallon-size ziplock plastic bag; add salt, black pepper, and 3 tablespoons of the pickle brine. Seal bag, and massage until chicken is evenly coated with brine mixture. Place in refrigerator; let marinate 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 8 minutes. Reduce heat to medium. Add onion; cook, stirring often, until onion softens, about 4 minutes. Add bell pepper, garlic, and chopped cherry peppers; cook, stirring often, until fragrant, about 1 minute.

Add wine to mixture in Dutch oven, scraping up any browned bits on bottom. Return mixture to a simmer over medium-high. Simmer, stirring occasionally, until reduced by half, about 2 minutes. Add tomatoes and juices, oregano, and chicken; bring to a boil over medium-high. Reduce to a gentle boil over medium; cook, stirring occasionally, until chicken is tender, tomato mixture thickens, and a spoon leaves a trail when dragged across bottom of pot, about 45 minutes, reducing heat as needed to maintain a gentle boil.

Meanwhile, cook pasta in boiling, salted water according to package directions for al dente. Drain pasta, reserving 1 cup cooking water. Set aside.

Add cream and remaining 3 tablespoons pickle brine to tomato mixture in Dutch oven; stir until combined. Stir in cooked pasta. Cook over low, stirring constantly, until tomato mixture clings to pasta, about 2 minutes. Season with additional salt to taste. Stir in reserved pasta cooking water as needed, ¼ cup at a time, to loosen sauce as desired. Divide pasta evenly among 8 shallow bowls. Garnish bowls with cheese, and serve.