Sprinkle the chicken with salt and pepper. Heat 1 tsp of the oil over medium high heat. Add the chicken and cook until lightly browned, about 3 minutes on each side. Transfer the chicken to a plate.
Add the remaining 1 tsp oil to the same skillet, then add the mushrooms; cook over medium high heat, stirring often, until the mushrooms are browned, about 4 minutes.
Whisk the Marsala, broth, and flour in a small bowl until blended Pour into the skillet along with the garlic. Cook, stirring constantly, until the mixture bubbles and thickens, about 1 minute. Return the chcicken to the skillet, turn to coat with the sauce, and sprinkle with the thyme. Reduce the heat and simmer, covered, turning the chicken occasionally, until the chicken is cooked through, about 5 mintues.
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