Put the polenta in a large saucepan with 1 cup of water. Whisk until smooth. Add 1 more cup of water and the 1 teaspoon of salt and bring to a boil, whisking constantly, over medium heat.

Turn the heat to medium-low and continue cooking and whisking frequently, adding water in small amounts, for about 30 minutes. The consistency should be thick yet somewhat loose.

Remove from the heat and stir in 2 tablespoons of butter and 1/2 cup of shredded Cheddar cheese. Add more cheese, if desired. Stir until smooth.

Add salt and freshly ground black pepper, as desired.

Meanwhile, prepare the sausage

Line a baking pan with foil; place a rack in the pan. Brush the rack with some vegetable oil.

Heat the broiler to high, or 500 F.

Arrange the sausages on the rack and broil 6 inches from the heat source for about 12 minutes. Turn the sausages over halfway through the cooking time.

When the polenta is ready, spoon some into a wide bowl or plate. Slice the sausages and arrange the slices over the polenta. Repeat with the remaining polenta and sausages.

Serve with simply dressed arugula or steamed spinach or broccoli.