Soft, fluffy sourdough discard hamburger buns sturdy enough for loaded burgers - no butter, milk, or egg needed, just pantry staples and your leftover discard.
Activate the yeast. Combine the lukewarm water, yeast, and 1 teaspoon sugar in a small bowl. Let stand for 5-10 minutes until foamy. If it doesn't foam, your water may have been too hot or too cold, or your yeast may be expired - start over with fresh yeast.
Make the dough. In a large mixing bowl, add the hot water. Stir in the salt and ½ cup of sugar until dissolved. Add the warm water, sourdough discard, oil, and vinegar, and stir. Add the prepared yeast mixture and stir until combined. Add 3 cups of flour and mix well. Gradually add the remaining flour, 1 cup at a time, until a soft, slightly sticky dough forms - it should start to pull away from the sides of the bowl but will still feel a little tacky to the touch. Depending on your sourdough discard and the humidity in your kitchen, you may need between 8 and 9 cups of flour total.
Knead. Turn the dough out onto a lightly floured surface and knead by hand for 8-10 minutes, or use a stand mixer fitted with a dough hook for 5-6 minutes, until the dough is smooth, elastic, and only slightly tacky. Dust your hands and the surface with a little extra flour as needed to keep the dough from sticking - you shouldn't need much.
First rise. Transfer the dough to a lightly greased bowl, turning once to coat the top with oil (this keeps a skin from forming). Cover and let rise in a warm place for about 1 hour, or until doubled in size.
Shape the buns. Pull a handful of dough and cut it free with a paring knife, then weigh it on a kitchen scale - about 95-100 grams per piece for large buns, or a slightly smaller portion for medium buns. To shape, pull the dough tight underneath itself, pinching the seam closed at the bottom, then turn the ball while holding the bottom seam in place until the top is smooth and taut. Place the shaped buns on parchment-lined baking sheets, leaving about 2-3 inches between each one to allow for spreading.
Second rise. Cover loosely with a clean kitchen towel and let rise for 45-60 minutes, until noticeably puffy and almost doubled - the buns should feel light and airy when gently touched, not dense. Watch the clock here: don't let them rise longer than an hour, or they'll spread out and go flat instead of staying domed.
Egg wash. Preheat your oven to 375°F. Whisk together the egg and 1 tablespoon water. Brush the tops gently with the egg wash and sprinkle with sesame seeds, if using.
Bake. Bake for 12-15 minutes, rotating the baking sheets halfway through, until the buns are golden brown and the internal temperature reaches 190-195°F.
Cool. Transfer to a wire rack and cool completely before slicing.
Notes
Flour. Add the final cups gradually - the amount needed varies with your discard's hydration and your kitchen's humidity.
Rise time. Keep the second rise to an hour or less. Going longer causes the buns to spread and go flat instead of holding a dome shape.
Storage. Store cooled buns at room temperature for 2-3 days, or freeze for up to 3 months.