In a microwave-safe bowl , combine the dry mustard and vinegar. Stir to combine, cover with plastic wrap and let sit overnight at room temperature.
The next day, add sugar and eggs and stir to mix well. Microwave on high for 6-7 minutes until the mixture bubbles and thickens, stirring every two minutes with a whisk to mix well.
Allow to cool while the kielbasa is grilled.
To grill the kielbasa, cut the kielbasa into thick slices on the bias.
On a medium grill, grill for just one to two minutes per side and serve immediately.
