Heat a large saucepan or Dutch oven to medium-high heat with the olive oil. Dry chicken pieces very well with paper towels and season on all sides with salt and pepper.
Add the chicken to the pan with 1 tablespoon of butter and sear for 3-4 minutes per side or until well browned. Work in batches adding 1 tablespoon of butter for the new batch and do not crowd the chicken. Place seared chicken pieces on a plate and sprinkle with a touch more salt.
Turn the heat to medium and add the remaining butter, peppers, and onions. Saute until soft (about 7-10 minutes) then add in the garlic and cook for 2 minutes more.
Next, add the tomato paste and a splash of water if it starts to burn. Cook the paste for 5 minutes, stirring frequently.
Next, add the white wine and chicken stock and turn the heat to high. Bring to a boil and scrape the bottom of the pan to remove all of the brown bits. Reduce the liquid by half then add in the plum tomatoes and return the chicken to the pan. Mix well and bring the sauce to a simmer.
Continue cooking at a low simmer until the chicken is very tender (about 30-40 minutes). Taste test the sauce and season with a bit of salt but not too much at this time since the salty capers and olives have yet to be added.
While the sauce is simmering, bring a large pot of salted water to boil (2 tablespoons salt per gallon of water) and cook the pappardelle until 1 minute less than al dente.
When the pasta is almost finished, remove the chicken pieces to a plate and cover with foil to keep warm. Add the capers and olives to the sauce and stir to combine.
Add the pasta to the pan. Toss or mix to coat and if needed add a splash of pasta water to loosen things up. Cook until just al dente then remove the pan from the heat. Taste test and adjust salt and pepper to taste and add the parsley.
Plate the pasta then serve with the chicken on top or on the side. Serve with grated parmesan. Enjoy!