In the bowl of a stand mixer fitted with the dough hook attachment (or a large mixing bowl), whisk together the warm milk, warm water, honey, and yeast. Let sit for 5–10 minutes until frothy.
Add the melted butter, sourdough discard, egg, salt, and 4 ¼ cups of flour. Mix on low speed for 2 minutes, or stir with a wooden spoon until a shaggy dough forms. If the dough is still very wet, slowly add up to ¼ cup more flour, just until it feels slightly tacky.
Kneed the dough.Using a Stand Mixer: Knead on low speed for 8–10 minutes, until smooth and elastic. If kneeding By Hand: Turn the dough out onto a floured surface and knead for 10–12 minutes. Use the heels of your hands and fold the dough over itself until it becomes soft and stretchy. Transfer the dough to a lightly oiled bowl, cover with plastic wrap or a damp tea towel, and let rise in a warm place for 1, or until doubled in size. Rise times may vary depending on the temperature of your kitchen.
Punch down the dough and turn it onto a lightly floured surface. Use a bench scraper to divide the dough into 10 equal pieces.Working one piece at a time:- Flatten into a rough 6×3-inch rectangle.- Roll up tightly from the long edge to form a log.- Pinch the seam and both ends to seal.- Place seam-side down on a parchment paper-lined baking sheet.Repeat with remaining dough, spacing buns apart to allow for rising. Place buns 1 inch apart for soft, pull-apart sides, or 2–3 inches apart for individual buns with golden edges. Cover the shaped buns with a towel or plastic wrap. Let rise for 30–45 minutes until noticeably puffy. Meanwhile, preheat the oven to 400°F.
Bake the buns for 12-15 minutes, or until the tops of the buns are golden and they reach an internal temperature of 190°F. Cool slightly before serving. For extra color, lightly brush the tops with milk or plant-based milk before baking. Once baked, brush the tops with melted butter while still warm for a soft, shiny finish. Perfect for regular ol’ hot dogs or even grilled sausages!