Drizzle 1 tablespoon of the oil in a 5 QT Dutch oven or large pot and bring to medium-high heat. Add the chicken to the hot oil and sprinkle with a bit of salt and pepper. Cook, stirring occasionally, for 6-8 minutes until cooked through and browned. If chicken releases liquid while cooking, drain and discard the liquid as needed so the chicken will sauté, not steam/boil. Add the garlic, the mushrooms, and 3 tablespoons of the marsala wine and stir together. Cook for 2-4 minutes, stirring occasionally, until the mushrooms are soft. Remove the chicken/mushroom mixture to a side plate and cover with aluminum foil to keep warm.
Lower the heat to medium and add the butter to the pot. When the butter is melted, whisk in the flour, salt and pepper until combined with the butter (will form a paste). Cook that for about a minute and then slowly whisk in the milk a little at a time to combine it smoothly with the flour mixture. When smooth, add the broth, the remaining marsala wine, the thyme and oregano, and whisk to combine, scraping up any remaining browned bits on the bottom of the pot.
Increase the heat to med-high and stir occasionally until boiling. Add the pasta and reduce the heat to med-low. Cook for 10-12 minutes uncovered, stirring every couple minutes to avoid sticking.
When the pasta is cooked and the sauce is thickened, stir in the Parmesan cheese. Reduce the heat to low and add the reserved chicken and mushrooms back to the pot. Stir everything together and cook for another couple minutes to ensure the reserved ingredients are warm. Sprinkle with parsley and serve.