Cook the pasta. Bring a large pot of water to a boil, season generously with about Kosher salt, then add pasta. Cook according to package directions to al dente. Before draining, reserve about 1 cup of the cooking liquid. Set aside.
Dredge the chicken. Pace chicken in a large bowl, then add ¼ cup flour, 1 tsp oregano, 1 tsp Kosher salt, and ¼ tsp black pepper. Mix well to thoroughly coat.
Brown the chicken. Heat a large pan over medium-high heat, then add 2 Tbsp olive oil and 2 Tbsp butter. Add dredged chicken, then cook for 3-5 minutes undisturbed. Stir well, then cook an additional 3-5 minutes, until golden brown on all sides. Transfer to a large plate.
Cook the mushrooms. Add another 2 Tbsp olive oil (or butter), then add sliced mushrooms. Cook 3-4 minutes undisturbed, until golden brown. Stir well, then cook an additional 3-4 minutes. Add sliced shallots, season with remaining ½ tsp Kosher salt, then stir and cook 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. Return the chicken (and any drippings) to the pan, bring to a boil, then lower the heat to a simmer. Allow the 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. Add ¼ cup Parmigiano Regianno cheese, then 2 Tbsp chopped parsley, then toss well. 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. Enjoy!