Put the potatoes in a large microwave-safe bowl and add 3 tablespoons water; cover with plastic wrap and microwave until tender, about 12 minutes.
Meanwhile, cut the kielbasa into 4 equal pieces; halve lengthwise. Melt 1/2 tablespoon butter in a large skillet over medium heat. Add the kielbasa and cook, turning occasionally, until browned and heated through, about 10 minutes. Transfer to a plate and cover loosely with foil to keep warm.
Increase the heat under the skillet to medium high; add the remaining 1 tablespoon butter. Add the broccoli and 1/4 cup water and cook, stirring, until crisp-tender, about 5 minutes. Add the sauerkraut and toss to warm through.
Drain the potatoes. Warm the milk in the microwave, about1 minute; add to the potatoes along with the mustard, 1/2 teaspoon salt, and pepper to taste. Coarsely mash with a potato masher or fork. Serve the kielbasa with the potatoes and the broccoli.