In a large, deep sauté pan, heat the oil and butter over medium-low heat. Add the onions and a pinch of salt. Slowly cook the onions for about 15 minutes, stirring occasionally, until they turn golden and begin to caramelize. Adjust the temperature and add more butter as needed.
Add the garlic to the pan and sauté for another minute until fragrant. Stir in the mushrooms and wine, and let the mixture simmer until nearly all of the liquid is absorbed, about 8-15 minutes. While the sauce is reducing, start preparing the pasta.
In a small bowl, whisk together 1 cup of cream with the flour until smooth and lump-free. Add the cream and flour mixture to the pan and stir well. Cook for a couple more minutes to thicken the sauce.
Add the Parmesan cheese and the remaining cream to the pan. Stir until the cheese has melted and the sauce is smooth. Then, add the remaining 2 tablespoons of butter and the cooked chicken, stirring until everything is well combined.
Cook the pasta until al dente. Before draining, check the Marsala sauce. If it's thinner than you'd like, add some of the pasta water, ½ cup at a time, but no more than 1 cup total. Cook the sauce for a few more minutes to thicken it to your desired consistency.
Drain the pasta and add it to the sauce, along with the fresh herbs. Season with salt and pepper to taste. Serve immediately, topped with freshly grated Parmesan cheese.