Radicchio and sausage balance each other nicely in this hearty dish. Round Chioggia radicchio is commonly available in supermarkets; if you can find it, try the slightly milder, spear-shaped Treviso variety. Add a crusty loaf of bread or some pasta tossed in olive oil, garlic, and chopped Italian parsley to complete the meal.
Melt the butter with 1 teaspoon olive oil in a large, heavy skillet over medium heat. Add the onions and cook until soft and golden brown, stirring often, about 25 minutes.
Add the radicchio and sauté until wilted, about 2 minutes. Add the balsamic vinegar and cook over medium-high heat until the liquid is absorbed, about 1 minute. Season to taste with salt and pepper. Keep the onion-radicchio mixture warm while preparing the sausages.
Heat the remaining 1 teaspoon olive oil in another large, heavy skillet over medium-high heat. Add the sausages and cook, turning frequently, until they are well-browned and heated through, about 5 minutes.
Serve the onion-radicchio mixture alongside the sausages.