Heat the oven to 450°; place oven rack 4-6 inches from the heating element; lightly grease a small baking sheet with butter.
Melt 2 tablespoons butter in a large saute pan over medium heat.
Add in garlic and thyme; saute until fragrant, about 1 minute.
Add all the mushrooms and season with salt and pepper; saute, stirring often, until tender, 6-8 minutes.
Meanwhile, slice the polenta into eight ½-inch thick rounds (you will have 1/3 of the roll left, save it for another use).
Arrange the polenta rounds on the prepared baking sheet and place on the oven rack; bake 5 minutes, then sprinkle about ½ tablespoon parmesan cheese on top of each slice and return to the oven until brown and crisp, 2-3 minutes.
When the mushrooms are tender, increase heat to high and stir in the marsala.
When is mixture is almost dry, add the broth; bring to a simmer, then decrease heat to med-low and stir in the cream.
Heat through, taste, and adjust w/ salt and pepper.
To serve, transfer the polenta slices to shallow bowls; ladle the ragout over the polenta and top with the remaining Parmesan and the chives.