Preheat oven to 400°F Spray a 3-quart casserole dish or clay cazuela with cooking spray (like PAM) and set aside. Spray a sheet pan with cooking spray and set that aside, too.
In a large deep skillet, heat 3 Tbsp olive oil over medium heat. Cook onion, garlic and rosemary, stirring occasionally until the onion is softened.
Add mushrooms and salt to taste, and cook over moderately high heat, stirring, for 10 minutes or until the liquid the mushrooms gives off is evaporated.
Stir in tomato paste and the wine, and boil until most of the liquid is evaporated.
In a small bowl stir the cornstarch into the broth, add the mixture and the Worcestershire sauce to the mushroom mixture and bring it to a boil, stirring. Reduce heat to simmer and cook 2 minutes. Season with salt and pepper to taste. Turn off the heat and set aside.
In a large, heavy saucepan, bring the water with 1 Tbsp olive oil to a boil and add 1 cup of the cornmeal, a little at a time, whisking constantly. Reduce heat to low, add the remaining 1 cup cornmeal in a slow stream, stirring constantly, and bring mixture to a boil. Remove pan from the heat and with a wooden spoon stir in the butter, 2/3 cup of the parmesan, the parsley, and salt and pepper to taste.
Spread 1/3 of the polenta on the sheet pan to 1/4 inch thick, and chill for 20 minutes, or until it is firm. While it is chilling, working quickly, spread half the remaining polenta in the prepared casserole dish or cazuela, top it with half the mushroom ragout, and top the ragout with the mozzarella. Spread the remaining polenta quickly over the mozzarella and top with the remaining ragout.
Invert the polenta sheet onto a work surface and with one or more cookie cutters, cut out as many pieces as possible. Arrange the pieces decoratively on the ragout and sprinkle with the remaining parmesan. (*The dish may be prepared up to this point and refrigerated, covered, for up to 2 days.).
Bake uncovered in the upper third of the oven for 30-40 minutes, or until the polenta pieces are golden.