In a small bowl, combine the lukewarm water, sugar, and yeast. Stir to dissolve, then cover and set aside for 10 minutes, or until the mixture is frothy and bubbly.

While the yeast mixture activates, whisk together the flour and salt in a large bowl to aerate. Set aside.

In a medium bowl, combine the butter and milk. Heat in the microwave just until the butter is melted. Set aside to cool slightly (this is important to avoid killing the yeast).

Once the yeast mixture is ready, add one egg and the melted butter and milk mixture. Stir to combine.

Add the wet ingredients to the flour mixture and stir until just combined, forming a sticky dough. Cover the dough and place it in a warm area to rise for 1–2 hours, or until it has doubled in size. (See notes for rising tips)

Transfer the dough to a lightly floured surface. Gently add small amounts of flour - only enough needed to handle the dough. Divide it into 10 even pieces, each weighing about 95–98 grams.

Flatten each dough ball, then roll it out into a hot dog bun shape.

Place the shaped buns onto a parchment-lined baking sheet. Cover and allow them to rise for about 1 hour.

Preheat the oven to 400°F (200°C). To make the egg wash, whisk together the second egg and 1 tablespoon of water. Brush the egg wash over each bun.

Bake for 12 minutes, or until the buns are golden brown.

Remove from the oven and let cool before slicing for your hot dogs.