Activate the yeast: Whisk the sugar, milk, and yeast in a large bowl. Cover and let the yeast bloom until foamy, at least 15 minutes.
Make the dough: Stir the sourdough discard into the yeast mixture until fully incorporated.
Whisk the salt and all-purpose flour together. Rub the unsalted butter into the flour until coarse crumbs form.
Create a well in the center and pour the yeast mixture in. Stir until no traces of dry flour remain.
Working inside the mixing bowl, knead the dough until it looks cohesive. If your dough is too sticky, cover the bowl with plastic wrap and let it rest for 15 minutes.
Knead the dough: Turn the dough out on a clean work surface. Knead the dough until smooth, about 5 to 10 minutes. Round the dough into a tight ball and transfer it to a lightly greased bowl.
Bulk ferment: Cover the bowl with plastic wrap and let the dough rise until doubled in volume, about 45 minutes to 1 hour (depending on the temperature of your kitchen).
Laminate the dough: Divide the dough into 5 even portions. Roll each portion of dough into a ball. Cover the dough with plastic wrap and let it rest for 10 minutes, this will make it easier to roll out.
Working with 1 portion of dough at a time, roll the dough into a 8-inch circle using a rolling pin.
Spread a tablespoon of butter on top of the disc, cover with plastic wrap.
Repeat with the remaining portions of dough, stacking each circle of dough on top of the previous one and spreading 1 tablespoon of butter on top, but do not spread butter on the top layer! (This will make rolling out the dough when shaping much easier.)
Pinch the edges of the dough together so the layers do not separate.
Transfer the layered dough to a plate and chill the dough for 1 hour.
Prep: Preheat the oven to 350 F and line a large baking sheet with parchment paper.
Shape the dough: Place the dough on a clean work surface. Use a rolling pin to roll it out into a large circle, about 15 to 18 inches, it should be quite thin (⅛-inch thick).
Using a pizza cutter or chef’s knife, divide the dough into 12 even segments (like a pizza).
Working with 1 triangle at a time, make a small slit in the middle of the base of the triangle. This will allow the dough to be curved into its iconic shape, so it’s important to make this cut.
Pull the two bottom corners outward, and roll the dough towards the tip.
Place the shaped dough on a large parchment-lined baking sheet and curve the ends so that the dough resembles a crescent shape (give the dough room to rise and spread). Repeat with the remaining portions of dough.
Final proof: Cover the baking sheet with plastic wrap and let the dough rise until puffy and has noticeably increased in volume, about 30 to 45 minutes.
Bake: Whisk the egg. Brush the tops of the dough with egg wash. Bake for 20 to 30 minutes or until the crescent rolls are golden brown.
Enjoy: These rolls are best served fresh out of the oven.
Store: Store any leftovers in an airtight container at room temperature for up to 3 days. Refresh the rolls in an oven toaster for 5 minutes.