1. To prepare polenta, bring 4 1/2 cups water to a boil in a medium saucepan. Gradually add polenta and 1 teaspoon salt, stirring constantly with a whisk. Reduce heat to low; cook 3 minutes or until thick, stirring frequently. Spoon polenta into a 13 x 9–inch baking pan coated with cooking spray, spreading evenly. Press plastic wrap onto surface of polenta; chill 1 hour or until firm.
2. Preheat broiler.
3. Invert polenta onto a baking sheet coated with cooking spray. Cut polenta into 36 rectangles (leave polenta on baking sheet). Broil 5 minutes. Turn polenta over. Brush polenta with 1 tablespoon butter. Broil 5 minutes. Sprinkle 1/2 cup shredded cheese evenly over rectangles; broil 5 minutes or until cheese melts.
4. To prepare topping, combine 1 cup boiling water and porcini mushrooms in a bowl; cover and let stand for 15 minutes. Drain; chop mushrooms.
5. Melt 1 tablespoon butter in a large nonstick skillet over medium heat. Add shallots to pan; cook 4 minutes or until tender, stirring frequently. Stir in red pepper; cook 30 seconds. Add porcini, shiitake, and cremini mushrooms; cook over medium-high heat 10 minutes or until liquid evaporates, stirring occasionally. Stir in vermouth, thyme, 1/4 teaspoon salt, and black pepper; cook 2 minutes, stirring frequently.
6. Spoon 1 teaspoon mushroom mixture on top of each polenta rectangle; top rectangles evenly with shaved cheese.