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Step 1
In the bowl of a stand mixer, dissolve the sugar into the warm water, sprinkle the yeast on top and let it bubble up, about 5 minutes. Pour the milk, melted butter and egg into the yeast mixture and whisk until smooth. Pour the flour on top. Knead with the dough hook attachment just until a shaggy dough forms, then stop. Allow the dough to relax, uncovered, for 20 minutes.
Step 2
Add the salt to the top of the dough, continue kneading on medium speed until the dough is smooth and elastic, about 8-10 minutes. Remove the bowl from the mixer, cover with plastic and allow to bulk ferment at room temperature for 1-1.5 hours or until nearly doubled in size.
Step 3
After the dough has risen, punch it down and divide it into 8 portions roughly 130-140 grams each. Shape each portion into a small puck shape and allow it to rest for 10 minutes. Using the tips of your fingers and pinching the corners of the dough shape it into a rectangle about 6 by 8 inches long. Place the filling on one side of the rectangle, then fold the dough over hamburger-style and seal the edges. Place the shaped Runza on a parchment paper-lined baking sheet and allow to proof until noticeably puffy, about 30-45 minutes.
Step 4
Preheat the oven to 375F.
Step 5
Once the runzas are puffy, bake them on the middle rack of the oven until the outside bread is golden brown, about 25 minutes. Carefully remove the runza from the oven and let them cool for 5-10 minutes before serving.
Step 6
Shape the ground beef into small patties and season with salt. In a large skillet over high heat, brown the ground beef over medium heat. When the patties take on color, break the meat up into crumbs with a cooking utensil, then add the chopped onion and garlic. Cook, stirring occasionally, until the beef is fully browned and the onions are soft, about 5 minutes. Add the sauerkraut, a pinch of salt and a few cracks of black pepper. Continue cooking until the pan looks dry and almost all the moisture has been cooked out of the vegetables, about 8 minutes.
Step 7
Drain off any excess liquid from the pan, then set the mixture aside to cool before using.