For the Dough: In a 4-cup liquid measuring cup or medium bowl, whisk milk, water, sugar, butter, and egg until smooth and sugar is dissolved.

In the bowl of a stand mixer fitted with a dough hook attachment, combine flour, yeast, and salt and mix on low speed until combined, about 20 seconds. With the mixer running on low speed, pour liquid mixture into the flour in a slow, steady stream. Continue mixing until a shaggy dough forms, stopping to scrape the sides of the bowl with a flexible spatula or dough scraper as needed, 2 to 3 minutes. If dough is excessively dry and dry flour remains, add water, 1 tablespoon at a time, mixing for 30 seconds to 1 minute after each addition, until all flour is hydrated and incorporated. Once no visible flour remains and dough is fairly cohesive, increase mixer speed to medium and knead until dough is smooth, elastic, and pulls away from sides of bowl, about 8 minutes.

Transfer dough onto a lightly floured surface and knead by hand to form a smooth ball, about 1 minute. Lightly grease a large bowl with nonstick cooking spray. Place dough seam-side down into bowl and cover tightly with plastic wrap. Let rise at room temperature until dough is doubled in size, 45 minutes to 1 hour.

For the Filling: While dough proofs, heat oil in a 12-inch cast iron skillet or straight-sided stainless steel sauté pan over medium-high heat. Add beef and onion and cook, breaking up meat and stirring often with a wooden spoon, until beef is browned, onion is just tender, and any moisture in the pan has cooked off, 12 to 15 minutes. Reduce heat to medium and stir in salt, pepper, paprika, garlic powder, and onion powder until well incorporated.

Add cabbage to pan a handful at a time, stirring between additions to combine. When all cabbage has been added, pour in water and stir to scrape up any browning on bottom of skillet. Continue cooking, stirring often, until liquid is evaporated and cabbage is wilted and tender with a slight bite, about 10 minutes. Remove from heat, season to taste, and let filling cool at room temperature, at least 20 minutes or up to 1 hour.

For the Bierocks: Once dough is proofed, adjust oven racks to upper-middle and lower-middle positions and preheat to 350°F (175ºC). Line two 13- by 18-inch rimmed baking sheets with parchment paper. Using your hands, gently press dough to deflate it and transfer to a lightly floured work surface. Using a bench scraper or sharp knife, quarter the dough, then cut each quarter in half so you have 8 pieces of dough weighing about 105g each. (See notes.) Shape each piece roughly into a ball and set aside. Cover loosely with a clean kitchen towel or plastic wrap to keep dough from drying on top.

Working with 1 dough ball at a time, use a rolling pin to shape into an 8-inch round, rolling from the center outward and lightly increasing pressure toward the edges so edges are slightly thinner than the middle. (Dough should be soft and pliant; if it recoils while rolling, let it rest for 5 minutes before continuing.) Place ¾ cup of cooled filling in a rounded mound in center, leaving a border of at least 1 inch around edges.

Gently bring edges of dough over filling; pleat edges of dough repeatedly, pinching the edge closed after each pleat until the entire bierock is sealed in a cinched purse shape. Pinch and twist top to seal, gently pressing down on the top to flatten slightly.

Place bierock, smooth-side down, into the cupped palm of your hand and, using your other hand, rotate bierock, putting gentle pressure on its sides as you turn, to shape into a rounded bun. Turn bierock over and place it seam-side down on parchment-lined baking sheet. Repeat with remaining dough, placing 4 evenly spaced bierocks on each pan.

Allow bierocks to rest until puffy and dough bounces back slightly when touched, about 15 minutes. Meanwhile, in a small bowl, whisk egg and water until smooth. Using a pastry brush, lightly brush top and sides of each bierock with egg wash.

Bake until tops are deep golden brown, 25 to 27 minutes, rotating the trays top-to-bottom and back-to-front halfway through baking. Transfer to a wire rack to cool for 5 minutes, then serve with mustard or other condiments of choice.