1. Bring 7 cups water to a boil in a large saucepan; add 4 teaspoons salt. Whisking constantly, add cornmeal a little at a time. Reduce heat to medium-low; cook, stirring frequently with a wooden spoon, until polenta pulls away from sides of pan and is very thick, about 25 minutes.

2. Heat oil in a medium skillet over medium-low heat until hot but not smoking. Add prosciutto and garlic. Cook, stirring, until prosciutto is slightly crisp, about 5 minutes; stir into polenta. Season with salt and pepper, if desired.

3. Pour polenta mixture into a 9-by-13-inch baking pan; smooth with a dampened spatula. Let cool at least 1 hour.

4. If not serving polenta immediately, cover with plastic wrap, and refrigerate until ready to use (or overnight).

5. Trim 1/4 inch from all sides of polenta. Cut polenta into 48 squares. Preheat broiler with rack 5 inches from heat. Lightly brush a metal baking pan with oil; heat under broiler 30 seconds. In batches, broil squares in pan, 2 minutes. Place cheese on each square; season with pepper. Broil until cheese is melted, about 1 1/2 minutes. Garnish with herb sprigs.