For polenta, in a medium saucepan bring 2 1/2 cups of the broth to boiling. In a small bowl stir together the remaining 1 cup broth and the cornmeal. Slowly add the cornmeal mixture to the boiling broth, stirring constantly. Cook and stir until mixture returns to boiling. Reduce heat to medium-low; cook 20 minutes or until mixture is thickened and creamy. Stir in cheese.
Meanwhile, in a large skillet cook mushrooms and shallots in hot oil and butter over medium heat 20 minutes or until tender, browned, and moisture has evaporated. Stir in chopped thyme, salt, and pepper. Serve over polenta and top with thyme sprigs.