In the bowl of a stand mixer, combine yeast, warm water, and 1/2 tsp sugar. Stir to combine and let sit for 5 minutes until frothy.
Add 1/4 cup sugar, melted butter, warm milk, 1 egg, salt, and 3 cups of flour to the yeast mixture. Using the dough hook, knead on low speed, adding the remaining 1 cup of flour as needed, until the dough pulls away from the sides of the bowl, about 6-8 minutes.
Shape the dough into a ball, place it in a greased bowl, cover with a damp cloth or plastic wrap, and let it rise for about 1 hour or until doubled in size.
Punch down the dough and transfer to a clean surface. Cut the dough in half and roll each portion into a 12-inch circle. Cut each circle into 12 wedges. Roll each wedge starting from the widest edge towards the point, shaping them into crescent rolls.
Place the rolls on a parchment-lined baking sheet. Cover with plastic wrap and let them rest for 30 minutes.
Preheat the oven to 375°F (190°C).
In a small bowl, whisk together 1 egg and 1 tbsp water. Brush the egg wash over the rolls.
Bake the rolls for 12-15 minutes, until golden brown and puffed up.
Remove from the oven and cool on a wire rack for a few minutes before serving.