Bloom the yeast: Whisk the sugar, milk and yeast in a large bowl. Cover the bowl with plastic wrap and let the yeast bloom until foamy.
226 grams milk, 28 grams cane sugar, 7 grams active-dry yeast
Make the dough: Stir the sourdough discard and egg into the yeast mixture.
1 large egg, 113 grams sourdough discard
Fold the flours and salt in using a stiff spatula or your hands. Bring the ingredients together until a shaggy dough forms and no traces of dry flour remain.
420 grams all-purpose flour, 60 grams whole wheat flour, 8 grams salt
Knead the butter into the dough a tablespoon at a time. The dough will feel gradually less greasy as the butter is absorbed.
56 grams unsalted butter
Cover the bowl with plastic wrap and let the dough rest for 15 minutes, this rest period will make the dough much easier to work with.
Knead the dough: Turn the dough out on a clean work surface, if your dough feels too sticky feel free to use a little bit of bench flour.
To knead the dough, hold the bottom of the dough with your fingertips, fold the dough over itself, then push it out using the heel of the palm of your hand. Repeat until the dough looks cohesive, smooth and passes the windowpane test.
First rise: Round the dough into a tight ball and place it in a clean, lightly-oiled bowl. Cover with plastic wrap and let the dough rise until doubled in volume, about 45 minutes to 1 hour.
Shape the dough: Punch the dough down to release the air. Divide the dough into 15 equal pieces. Round each portion into a tight ball, and arrange the dough balls on a parchment-lined baking sheet in a 5 by 3 pattern.
Final proof: Cover the baking sheet with plastic wrap and let the dough rise until the rolls fill the baking pan most of the way, about 30 to 45 minutes.
Prep: When your rolls are close to being fully proofed, preheat your oven to 350 F.
Egg wash: Beat the egg and pinch of salt together. Using a pastry brush, brush the tops of your dough with egg wash.
1 egg, Pinch of salt
Bake: Bake the rolls for 20 to 25 minutes or until the rolls are golden brown and register at at least 190 F when probed with an instant-read thermometer.
Store: Store any leftover rolls in an airtight container at room temperature for up to 3 days.