Slice the chicken breasts horizontally into thin cutlets. Place the chicken in a plastic bag or between sheets of parchment paper and gently pound to about 1/4 inch thick.
Combine the flour, salt, and pepper in a shallow bowl. Lightly dredge each piece of chicken in the flour and shake off the excess. Place the floured chicken on a rack over a sheet pan to keep it dry while you prep the rest.
Heat 1 tablespoon butter and 1 tablespoon olive oil in a large stainless steel skillet over medium high to high heat. Working in batches, sear the chicken for 2 to 3 minutes per side, until golden brown. The chicken does not need to be completely cooked yet. Transfer to a platter and continue with the remaining chicken, adding additional butter or olive oil as needed. If the butter begins to burn, carefully wipe the skillet clean and start fresh before continuing.
Add 1 tablespoon butter to the skillet, followed by the mushrooms. Cook over fairly high heat for 8 to 10 minutes, stirring occasionally, until the mushrooms are deeply browned. Add the onion and 1/2 teaspoon salt and cook until softened. Add the garlic and fresh thyme and cook for another minute.
Pour in the Marsala and scrape up the browned bits from the bottom of the skillet. Allow it to simmer for a few minutes, then add the chicken stock and heavy cream. Bring to a simmer and cook uncovered, stirring occasionally, for about 30 minutes or longer, until the sauce has reduced significantly and become rich and flavorful. Taste and add additional salt if needed.
Return the chicken to the skillet and simmer for about 10 minutes, spooning the sauce over the chicken occasionally, until the chicken is cooked through and the sauce has thickened slightly.
Serve the chicken with buttered vermicelli pasta and spoon plenty of mushrooms and Marsala sauce over both the chicken and pasta.