Melt butter with 1 teaspoon olive oil in heavy large skillet over medium heat. Add red onions and cook until soft and golden brown, stirring often, about 25 minutes. Add radicchio and sauté until wilted, about 2 minutes. Add balsamic vinegar and cook over medium-high heat until liquid is absorbed, about 1 minute. Season to taste with salt and pepper. Keep onion-radicchio mixture warm while preparing sausages.
Heat remaining 1 teaspoon olive oil in another heavy large skillet over medium-high heat. Add chicken-apple sausages and cook until sausages are well browned and heated through, turning frequently, about 5 minutes.
Serve onion-radicchio mixture alongside sausages, perhaps with a small side of pasta or crusty bread.