Season the chicken breasts on both sides with salt and pepper. 3/4 to 1 teaspoon of kosher salt per pound of meat is a good guideline if you are not seasoning your meat by eye. Keep in mind that table salt is typically "saltier" than kosher salt.
Place the flour in a bowl and coat the chicken breasts evenly, shaking off any excess.
Over medium high heat, melt the butter in a large skillet or braiser (I used a 14" Le Creuset braiser pan), then brown the flour-dredged chicken for about 5 minutes each side, until 150F inside. We want to mostly cook them through, but they don't need to be completely cooked at this point.
Remove the chicken to a plate, then add the mushrooms, thyme, and Marsala wine. Cook, stirring frequently, for about 3 minutes, until the wine has mostly cooked off.
Add the garlic, and stir for 1 minute.
Add the heavy cream and chicken broth. Bring to a simmer, and cook for 2 minutes.
Add the chicken back to the pan, including any juices on the plate. Simmer for about 5 more minutes, basting the chicken in the sauce, until the sauce has thickened and reduced down, and the meat is fully cooked with an internal temperature of 165F. Garnish with extra thyme leaves, then serve and enjoy!
*Sometimes chicken breasts are absolutely enormous, and those will not work well here. When in doubt, cut the chicken breasts into cutlets or pound them out to an even 1/2" thickness. If the chicken is too thick or the pieces are too large, it will take too long to cook through.**Try to use a dry Marsala instead of a sweet variety. However, if you can only find sweet, that should still work sufficiently.Wine substitutes: It won't be the same, but this dish would still taste good with other options such as Madeira, Sherry, or Port. You can also try a half amount of Brandy, or other types of dry white wine.Mushroom substitutes: You may also try button, Hen of the Woods, Chanterelle, Porcini, or other mushrooms that saute well.Storing Leftovers: Store in an airtight container in the fridge for up to 5 days.Freezing: I never like freezing dishes with dairy because the sauce inevitably curdles. Technically, it will keep in the freezer for 2 months, but I don't recommend freezing.Reheating: I reheat this dish in the microwave until warmed through, but you can also reheat on the stovetop. Gently simmer for about 5 minutes, making sure not to put the heat too high, or the sauce will curdle.