Polenta and Sausage Stuffing

Butter a shallow baking pan (15 by 10 inches). Bring 6 cups water with salt to a boil in a 4-quart heavy pot, then add polenta in a stream, stirring with a long-handled spoon, and simmer, stirring constantly, 5 minutes (polenta will be very thick). Add 3 tablespoons butter and stir until butter is incorporated. Spread polenta in buttered pan and chill, uncovered, until firm, about 15 minutes.
While polenta is chilling, cook sausage in 1 tablespoon oil in a 10-inch heavy skillet over moderately high heat, breaking up lumps, until no longer pink, about 3 minutes, then transfer with a slotted spoon to a large bowl. Add onion to skillet and cook over moderate heat, stirring occasionally, until browned, about 3 minutes; add garlic and cook, stirring, until fragrant, about 30 seconds. Add remaining 1/2 cup water and stir up brown bits from bottom of skillet, then add onion mixture and chicken broth to sausage.
Preheat broiler. Melt 1 tablespoon butter with remaining tablespoon oil in a small saucepan. Invert polenta onto a large cutting board, then cut half of it into 1/2-inch cubes (reserve remainder) and toss with butter mixture in a medium bowl. Return polenta cubes to baking pan, spreading evenly, and broil 3 to 4 inches from heat until golden brown in patches, 8 to 12 minutes.
Put oven rack in upper third of oven and preheat oven to 450°F. Butter a 13- by 9-inch baking dish (3-quart capacity).
Coarsely mash remaining polenta with a potato masher and add to sausage mixture. Add broiled polenta cubes, 1/2 cup cheese, parsley, and pepper to taste, and toss until combined well. Transfer to baking dish. Sprinkle top of stuffing with remaining 1/2 cup cheese and bake, covered tightly with a sheet of buttered foil (buttered side down), until heated through, about 20 minutes. Remove foil and bake until top is lightly browned, 10 to 15 minutes more.