The filling for the buns can be made (and refrigerated) up to one day in advance. For quicker assembly, portion the filling into 12 (1/2-cup) mounds before rolling out the dough. Eat warm or at room temperature.
1. FOR THE FILLING: Heat 2 teaspoons oil in Dutch oven over medium-high heat until just smoking. Add beef, 1 teaspoon salt, and 1 teaspoon pepper and cook, breaking up pieces with spoon, until liquid evaporates and meat begins to sizzle, 10 to 15 minutes. Drain meat in colander and set aside.
2. Add remaining 2 teaspoons oil to now-empty Dutch oven and heat over medium-high heat until shimmering. Add onion and cook, stirring occasionally, until just beginning to brown, about 2 minutes. Stir in drained beef, cabbage, and chicken broth. Bring to simmer, cover, and cook until cabbage is tender, about 5 minutes. Uncover and continue to cook until liquid evaporates and mixture begins to sizzle, 5 to 7 minutes. Season with salt and pepper to taste. Transfer to bowl and let cool 15 minutes. Stir in Colby Jack and let cool to room temperature. Refrigerate until ready to assemble bierocks.
3. FOR THE DOUGH: Meanwhile, grease large bowl. Combine milk, water, butter, and sugar in 2-cup liquid measuring cup. Microwave until temperature registers 110 degrees, 1 to 2 minutes. Using stand mixer fitted with dough hook, mix 3½ cups flour, yeast, and salt on low speed until combined, about 30 seconds. With mixer running, slowly add milk mixture and single beaten egg until incorporated and dough comes together, about 3 minutes. Increase speed to medium-low and continue to mix until dough is smooth and pulls away from sides of bowl, about 8 minutes. If after 4 minutes dough looks wet, add remaining ¼ cup flour, 1 tablespoon at a time, until dough clears sides of bowl but sticks to bottom. Transfer dough to greased bowl, cover with plastic wrap, and let rise in warm place until doubled in size, 50 to 60 minutes.
4. Adjust oven racks to upper-middle and lower-middle positions and heat oven to 375 degrees. Line 2 rimmed baking sheets with parchment paper. Transfer dough to lightly floured counter and divide into 12 equal (2¾-ounce) portions. Form dough pieces into balls, cover with plastic, and let rest for 5 minutes.
5. Working with 1 dough ball at a time (keep remaining balls covered with plastic), roll out into 5½-inch round with rolling pin. Transfer ½ cup lightly packed filling (3 ounces) to center of round. Use hands to press filling into compact mound. Stretch edges of dough over filling and pinch together to form bun. Arrange 6 bierocks, seam side down, on each prepared baking sheet, 2 inches apart. Cover with plastic and let rise until doubled in size, 45 to 60 minutes.
6. Discard plastic, brush bierocks with egg wash, and bake until golden brown, 22 to 25 minutes, switching and rotating sheets halfway through baking. Transfer sheets to wire rack and let cool for 15 minutes. Serve.
TO MAKE AHEAD: Baked, cooled bierocks can be tightly wrapped in aluminum foil, placed in zipper-lock bag, and refrigerated for up to 3 days or frozen for up to 1 month. To reheat, adjust oven rack to middle position and heat oven to 350 degrees. Place foil-wrapped bierocks directly on oven rack until heated through, about 10 minutes if refrigerated or 60 minutes if frozen.