Cook the pasta. Bring a large pot of water to a rapid boil, then season generously with about 2 Tbsp Kosher salt. Add pasta, then cook according to package directions, until al dente. Before draining, reserve about 1 cup of the cooking liquid. Set aside.

Dredge the chicken. Cut chicken thighs into roughly 1" pieces, then place in a large bowl. To this, add ¼ cup flour, 1 tsp oregano, 1 tsp Kosher salt, and ¼ tsp black pepper. Mix well to thoroughly coat the chicken on all sides.

Brown the chicken. Heat a large skillet or braiser over medium-high heat, then add 2 Tbsp extra virgin olive oil and 2 Tbsp unsalted butter. Add seasoned chicken, then cook for 3-5 minutes undisturbed. Stir well, then cook an additional 3-5 minutes, or until golden brown on all sides. Transfer chicken to a large plate.

Cook the mushrooms. Add another 2 Tbsp extra virgin olive oil (or butter), then add sliced mushrooms. Cook 3-4 minutes undisturbed -- this allows them to take turn golden brown. Stir, then cook an additional 3-4 minutes until evenly browned. Add sliced shallots, season with remaining ½ tsp Kosher salt, then stir and cook until translucent, about 2 minutes more.

Make the chicken marsala sauce. Pour in ¾ cup marsala wine, ¼ cup sherry vinegar, and 2 Tbsp balsamic vinegar. Use a spatula to scrape any brown bits off the bottom of the skillet. Return the chicken (and any drippings) to the pan. Bring to a boil, then lower heat to a simmer. Allow sauce to reduce and chicken to finish cooking through, about 3 minutes.

Add the pasta. Add cooked pasta to the pan, then pour in about ½ cup reserved pasta water. Grate ¼ cup Parmigiano Regianno cheese on top, then sprinkle with 2 Tbsp chopped parsley. Toss well, then taste and adjust seasoning as needed. If you want more of a sauce, add additional reserved pasta water. Serve immediately with additional Parmesan cheese and freshly grated black pepper.