In the bowl of a stand mixer fitted with the hook attachment, combine the warm milk (or water), yeast, and sugar. Give it a quick stir to combine then let sit for 5 minutes, until very foamy.
Whisk in the salt and egg.
Add the flour and knead until flour is fully absorbed.
On LOW speed, add the cubed softened butter one tablespoon at a time. Knead an additional 1 to 2 minutes, until a soft and pliable dough.
Cover the bowl with plastic wrap or a damp kitchen towel to rise 45 minutes to 1 hour, until dough has doubled in size.
Preheat oven to 375F. Line either a sheet pan or 9×13″ baking pan with parchment paper. For wider hotdog buns, use the sheet pan. For taller, use the 9×13″ baking pan.
Transfer the dough onto a clean surface. Divide into 10 equal sized pieces. Roll each into a tight log that’s 4″ to 5″ long.
Evenly distribute on the prepared baking pan. Press down on each hot dog bun to be around 1″ tall.
Cover to rise 30 to 45 minutes, until doubled in size.
Brush the very tops with the beaten egg, then bake for 10 to 15 minutes, until deeply golden on top. Rotate the pan halfway through baking.
Immediately brush on 1 to 2 tablespoons melted butter after baking. Let rest at least 15 minutes before slicing to enjoy.