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Add the remaining 2 tablespoons oil to the same skillet over medium-high heat and saute the mushrooms until softened, 2 minutes. Add the wine and cook 1 minute before adding the beef broth and thyme. Bring the sauce to a simmer and let reduce by three-quarters, about 15 minutes. The sauce should have thickened and turned a rich dark brown. Just as the sauce is almost done, return the chicken to the pan and finish cooking through to 165 degrees F, about 5 minutes. Season with salt and pepper.
Serve the chicken with the mushrooms and Madeira sauce and garnish with fresh thyme.