Season the chicken on both sides with salt and pepper. Coat in flour, shaking off excess. (Reserve about 1 tablespoon of flour per cup of broth, or more if needed)
Heat oil in a nonstick pan over medium heat. Cook chicken on both sides to golden brown, then remove from the pan and set aside.
Add shallots, mushrooms, and thyme to pan. Sauté until mushrooms are soft and beginning to brown, about 5 minutes.
Add Marsala wine and simmer until the liquid reduces in volume, about 2 minutes. Add the broth and the chicken back into the pan; bring to a simmer.
Dissolve the reserved flour in an equal amount of cold water. Mix into a paste so there are no lumps.
Slowly pour and stir in a little of the flour/water mixture at a time into the pan and let it simmer to thicken. Continue adding a little of the flour/water mixture as needed to reach a thick smooth consistency of your liking. Taste and adjust the seasoning with salt and pepper if needed. Remove the thyme sprigs before serving.
Serve with dinner rolls.