Heat the EVOO in a deep skillet over medium-high heat, add the butter and melt into the oil. Add the mushrooms and brown well. Add the garlic, shallots, herbs and some salt and pepper, and cook for 2 to 3 minutes more. Then add the tomato paste, stir 1 minute more. Add the marsala, stir. Add the passata, stock and cream, and combine. Add the chicken and simmer over low heat to thicken. Adjust seasoning. Cool and store for a make-ahead meal.
Bring a pot of water to boil for the pasta, salt the water and cook the rigatoni to al dente. Reserve a cup starchy water, then drain. Toss the pasta with the sauce and add the starchy water as needed if the mixture seems dry or tight. Serve topped with Parmigiano-Reggiano.