Preheat broiler to high or oven to highest heat. Place peppers on foil-lined baking sheet and roast, charring skin evenly. Cool peppers in covered bowl to handle, then peel, seed, and chop.
Meanwhile, heat a large pot of water to boil for pasta, 5 to 6 quarts.
Heat a large skillet over medium heat with EVOO, 3 turns of the pan, add onions and season with salt and oregano, add fresh chilies or Calabrian paste and cover pan, soften 6 to 7 minutes, stirring occasionally. Add garlic, stir, add wine or vodka and let it absorb, add tomatoes and basil and reduce heat to low simmer, add cream, then stir in roasted peppers and chicken.
Cook pasta 1 minute less than package directions and reserve half a mug of starchy water.
Toss pasta with sauce and cheese and add starchy water as needed to combine. Serve family-style or in shallow bowls topped with chopped hot pickled peppers.
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