Prep & Poach Chicken: Place chicken, ¼ cup cherry peppers, half the garlic, red bell pepper, and oregano in a cheesecloth pouch. Simmer in salted boiling water for 5 min. Turn off heat, add chicken, cover, and poach for 6 min. Remove chicken, cool, and shred. Keep broth simmering.
Cook Pasta: In the flavorful broth, cook rigatoni until almost al dente (2 mins less than package time). Reserve 2 cups pasta water, then drain.
Brown Mushrooms: In a Dutch oven, heat 2 tbsp oil over high heat. Brown mushrooms (8-10 mins), then remove.
Build Sauce Base: Add remaining oil to the pot. Cook onions until soft (5 mins). Add remaining garlic and cook 1 min. Stir in pepper flakes and tomato paste; cook 2 mins. Deglaze with sherry.
Simmer: Add roasted peppers, canned tomatoes, basil, 1 tsp salt, mushrooms, and shredded chicken. Simmer on low.
Finish the Dish: Add par cooked pasta, cream, butter, and 1 ½ cups Romano cheese to the sauce. Stir in reserved pasta water until sauce is creamy. Cook 2-3 mins until pasta is tender.
Serve: Garnish with remaining cherry peppers, basil, and Romano cheese.