Mix the dough. Combine the sourdough starter, flour, water, sugar, and oil in a large mixing bowl. Stir until a shaggy dough forms. Then, cover the bowl with a clean kitchen towel and let it rest for 30 minutes to allow the gluten to develop (autolyze).
Knead the dough. Add the salt to the dough and knead for 5–7 minutes. You can knead by hand, use a stand mixer with a dough hook, or a bread machine on the dough cycle. The dough should be soft, smooth, and slightly tacky to the touch. If it sticks excessively, add flour a teaspoon at a time. If it feels stiff or dry, add water a teaspoon at a time.
First rise. Transfer the dough to a clean, lightly oiled bowl. Cover and let rise in a warm place for 3–4 hours or until the dough is puffy. It may not double in size, but it should look noticeably risen.
Shape the buns. Turn the dough out onto a lightly floured surface. Then, divide it into 10 equal pieces (or 15 for slider-sized buns). Next, roll each piece into a smooth ball by cupping your hand around the dough and rolling it gently against the countertop. Arrange the balls on a parchment-lined baking sheet, spaced evenly apart.
Final proof. Cover the buns with heavily greased plastic wrap or a damp tea towel to prevent drying out. Let them rise in a warm spot for 3–4 hours or overnight in the refrigerator until they appear puffy and slightly increased in size.
Prepare for baking. Preheat your oven to 400°F (200°C), positioning a rack in the center. Then, place an empty-rimmed baking sheet in the oven while preheating for a soft crust with good rise. When ready to bake, brush the tops of the buns with the beaten egg and sprinkle with sesame seeds.
Bake the buns. Pour 1 cup of water into the preheated baking sheet to create steam, then quickly place the buns in the oven. Bake for 15–18 minutes until the buns are golden brown. The larger buns may need the full-time; smaller buns may finish earlier.
Cool completely. Transfer the buns to a wire rack and allow them to cool completely before slicing and serving. This helps them retain their structure and softness.