Preheat oven to 400ºF.
Place a large skillet over medium-high heat with 1 turn of the pan of EVOO, about 1 tablespoon. While the pan is heating up, cut a slit in each of the kielbasa pieces so that you can open them up like a book. Place them cut side down into the pan and cook until they crisp up, 4-5 minutes (you'll have to do this in batches).
While the kielbasa are crisping up, place a medium-size skillet over medium-high heat with 1 turn of the pan of EVOO, about 1 tablespoon. Add the onion, mustard and about 2 tablespoons of vinegar to the pan with some salt and pepper. Cook the onion until tender, 5-6 minutes.