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.
Stuffing, without parsley and cheese, can be prepared (but not baked) 1 day ahead and chilled, covered. Bring to room temperature; stir in parsley before proceeding.