Step 1

Heat a large skillet over medium-high; melt 2 tablespoons butter. Pierce sausages with a fork; add to skillet. Cook until browned in places, 4 to 5 minutes. Transfer to a plate. Meanwhile, cut vegetables into 1/2-inch slices; add to skillet. Season with salt and pepper; cook, stirring, until browned in places, 4 to 5 minutes. Stir in tomato paste; cook 1 minute. Return sausages to skillet with 3/4 cup broth; bring to a boil. Cover, reduce heat, and simmer 4 to 5 minutes. Uncover; simmer until slightly thickened, 1 to 2 minutes more.

Step 2

In a saucepan, combine polenta, 2 cups broth, and 1 cup water; bring to a boil. Reduce heat to medium; cook, stirring constantly, until thick and smooth, 1 to 2 minutes. Remove from heat. Stir in 2 tablespoons butter and cheese; season with salt and pepper. Serve, with ragu, parsley, and more cheese and pepper.