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Whisk together flour, yeast, sugar, and salt in a large bowl.
Heat 3 tablespoons of the butter and milk in a small saucepan until butter has just melted. Remove from heat to cool slightly.
Whisk together warm water and one egg in a medium bowl. Add warm milk and butter mixture and whisk until combined. Pour over flour mixture and stir with a rubber spatula or wooden spoon until no dry flour remains and you have a sticky, shaggy dough. Cover bowl tightly with a lid or plastic wrap and let stand in a warm place 2 to 4 hours until dough has doubled in size. Or tightly cover and chill for 12 to 24 hours to rise slowly in the refrigerator. (My preferred way, for the best flavor and texture. I typically give it a head start at room temperature before refrigerating. Just enough to let the dough begin to bubble and rise, 30-60 minutes depending on how warm the kitchen is.)
Line a rimmed baking sheet with parchment paper.
Turn risen dough out onto a lightly floured work surface and gather dough into a ball with lightly floured hands, then gently pat into a rectangle. With a sharp knife or dough scraper, divide dough into 10 equal pieces (use a kitchen scale for perfectly even hot dog buns). Shape each piece into a ball by first tucking edges under then pinching together to seal. With a rolling pin, roll one ball into a rectangle about 6 inches long. Fold rectangle into thirds as you would a letter, folding one long side to the center and folding the opposite long side over the top. Turn the dough seam-side down and gently fold the ends under to seal. Repeat with remaining dough balls, placing each bun about 1/2-inch apart on prepared baking sheet in 2 rows of 5 side by side on the baking sheet (see image). Cover loosely with a clean kitchen towel and let rise until puffy and light to the touch, about 1 hour or up to 2 hours if dough is chilled.
When ready to bake, preheat oven to 375˚F.
Whisk together remaining egg with 1 tablespoon water and brush over buns with a pastry brush. Bake 12-15 minutes or until tops are golden brown, rotating sheet halfway through baking time.
When buns are nearly finished baking, melt the remaining 2 tablespoons of butter. Once you've removed buns from the oven, brush the tops with melted butter while they're still very hot. Transfer to a wire rack to cool completely.