Feed your starter - Add equal parts flour and water to your starter and stash it in a warm place until it is alive and bubbly.
Mix the dough and let it rest - Place the bowl of your stand mixer on your digital scale (set to grams), and tare out. Measure out the starter, then add the sugar, and water, and stir with a big spoon until the sugar has dissolved. Stir in the oil. Tare out again, and measure out the flour, then add the salt. Mix with your spoon (or by hand) until you have a shaggy dough. Cover the bowl and let it rest for about an hour.
Knead the dough - Using the dough hook, knead the dough for about 10 minutes, until it is smooth and supple. Cover the bowl again and let it rest for another couple of hours, beginning the fermentation process.
Shaping the rolls - Tip the dough out onto a clean surface, and cut it into equal pieces (e.g., 4 pieces for hoagie rolls, 8 pieces for hamburger rolls, etc.). Pull the dough into a rectangle or a square, and fold or roll the dough into the desired shape. Place the rolls on a Silpat lined baking sheet and cover with a wet towel. Let them proof for as long as it takes for them to become soft and puffy, about 4-5 hours.
Bake - When you're ready to bake, preheat the oven to 390℉. Brush the rolls with an egg wash and slide into the oven for 5 minutes. If you intend to score them, open the oven at the 5-minute mark and score the top of the rolls. Continue baking for another 15-20 minutes, until the internal temperature is at least 200°.
Cool - Remove the rolls from the oven and place them on a wire rack. Brush with butter to ensure that the outside of the rolls are soft. Allow the rolls to cool.