Proof your yeast. Combine yeast into warmed milk with a pinch of sugar. allow to sit for a few minutes.
In the bowl of a stand mixer with a dough hook attachment, whisk together the cups of flour, sugar, salt, and milk with yeast proofed in it. Add the sourdough discard, egg, and room temperature butter.
Mix on low speed until a shaggy dough forms, scraping the sides of the bowl as needed. Continue kneading for about 5–7 minutes until smooth and elastic.
Place dough into a greased bowl, cover with plastic wrap, and let the dough rise in a warm place until doubled, about 30–45 minutes.
Turn out onto a lightly floured surface. Flatten out with your hands to create a small rectangle.
Divide into 8 equal pieces.
Shape each piece of dough into smooth shaped rolls.
Arrange in a greased baking pan or parchment-lined baking sheet. Cover and let puff for 30 minutes.
Preheat oven to 375°F (190°C). Brush tops of the rolls with egg wash. Bake until golden brown—18-20 minutes. Cool slightly on a wire rack.
Store cooled rolls in an airtight container at room temperature for up to 3 days, or in a freezer bag for longer storage.