Heat oil in a deep 12-inch 5-quart saute pan over medium-high heat. Brown kielbasa coins for 5 to 6 minutes, turning once, then transfer to a plate.
Lower the heat to medium and melt butter in the same pan. Cook onion and cabbage with salt and pepper for 10 to 12 minutes, stirring often, until softened with a few browned edges.
Pour in broth and bring to a simmer. Stir in dry egg noodles, cover loosely and cook over medium-low heat for 8 to 10 minutes, lifting and stirring every 2 minutes until noodles are tender.
Add reserved water 2 tablespoons at a time only if the pan dries before the noodles finish. Fold kielbasa back in and heat for 2 minutes.
Remove from the heat and stir in vinegar. Divide into 6 portions while the noodles and cabbage are still loose and glossy.