Char red peppers over an open flame or under hot broiler. If broiling, leave oven door ajar to allow some of the steam to escape. Place charred peppers in a bowl, cover with plastic wrap and cool. Peel and seed then quarter lengthwise. Slice and reserve.
Bring a large pot of water to a boil for the pasta.
Heat 1 tablespoon EVOO, a turn of the pan, in a skillet over medium-high heat. Season chicken liberally with salt and pepper then add chicken pieces to the skillet in a single layer. Brown evenly on both sides, 5-6 minutes total, then remove to plate.
To the skillet, add remaining EVOO, 3 turns of the pan, then the mushrooms. Brown mushrooms 5-6 minutes then add onions, garlic, cubanelle and chili peppers. Season with salt and pepper, and saut to soften, 6-7 minutes more. Deglaze pan with Marsala or sherry, add tomatoes and break them up with a spoon or masher. Reduce heat to medium-low to simmer. Add torn basil leaves, reserved sliced roasted peppers and chicken back to the skillet and stir in half-n-half or cream. Let sauce thicken while the pasta cooks
When water is boiling, add salt and rigatoni. Cook pasta to al dente. Reserve a cup of starchy cooking liquid to toss with pasta sauce just before draining.
Toss chicken with "riggies" and use a splash or 2 of starchy water as needed to combine. Toss in a little cheese, serve in shallow bowls and pass more cheese at table.
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