Bring a large pot of water to a boil; add salt and then the rigatoni (don’t forget—a couple of ladles of cooking water will be added to the sauce just before the pasta is drained); cook to al dente.
Meanwhile, heat a large skillet over med-high heat; add in 2 tablespoons olive oil and 1 ½ tablespoons butter.
When the butter melts, add chicken to the skillet; season with salt and pepper; brown for 2-3 minutes on each side.
Transfer chicken to a plate; add 1 tablespoons olive oil and remaining butter to skillet; then the mushrooms and garlic; cook 10-15 minutes until the mushrooms are tender; salt and pepper the mushrooms after they brown (if you salt them when they are first added to the skillet, the salt will draw out the liquids and slow the browning process).
Add in the hot peppers and a splash of the pickling liquid to the skillet.
Add in the white wine and scrape up the pan drippings with a wooden spoon.
Cook the wine down for a minute, then slide the chicken back into the pan.
Cook together for another couple of minutes to finish cooking the chicken.
Right before you drain the pasta, add 2 ladles of starchy water to the skillet; drain the rigatoni well and add it to the skillet.
Turn off the heat and toss the chicken, mushrooms, and pasta together for a minute or two; sprinkle with Parmigiano-Reggiano cheese as you go, to allow the pasta to soak up the sauce and flavors.
Garnish with chopped parsley and pass extra cheese and crusty bread at the table.