Pound the chicken breast to an even 1/2-inch thickness and sprinkle with 1/4 teaspoon of the salt, 1/4 teaspoon of the dried thyme and the pepper.
Heat 1 teaspoon of olive oil in a large nonstick skillet over medium-high heat. Working in batches if necessary, add the chicken 9 but don't overcrowd the pan) and cook until browned and cooked through, 3-5 minutes on each side.
Remove from the pan and place on a plate.
Heat the remaining 1 teaspoon of oil in the skillet. Add the mushrooms and remaining salt and thyme. Cook, stirring occasionally, until the mushrooms are slightly browned, 4-5 minutes.
Add the shallots and cook, stirring often until the shallots start to soften, about 2 minutes. Add the garlic and stir for another minute.
Add the marsala and broth and cook until the sauce thickens a little, about 1 minute. Add the chicken back to the skillet and cook until heated through, about 2 more minutes.