Cut kielbasa on an angle, into 3- to 4-inch pieces. Warm oil in a large skillet over medium-high heat; add onion and caraway seeds. Cook, stirring often, until onion is slightly softened, about 3 minutes. Add sausage and cook, turning often, until lightly browned, about 5 minutes.
Place sauerkraut on top of sausages and pour in broth. Season lightly with pepper. Cover and cook until warmed through, about 10 minutes. (If sauerkraut seems too wet, boil rapidly for a few minutes. As it boils and liquid evaporates, flavors will also be concentrated.)
Serve hot with warmed applesauce on side or with mustard, if desired.