3 tablespoons extra-virgin olive oil, plus more for brushing
2 1/4 cups all-purpose flour, plus more for dusting
1 teaspoon kosher salt
1 large egg white
Grated parmesan, for topping
Combine 3/4 cup warm water (105 degrees F to 110 degrees F) and the sugar in a food processor. Sprinkle the yeast on the surface and set aside until foamy, about 5 minutes. Add the olive oil and pulse once. Add the flour and salt; pulse to form a smooth dough, about 2 minutes. Brush a large bowl with olive oil. Transfer the dough to the bowl and form into a ball. Cover with plastic wrap and set aside in a warm place until doubled in size, about 1 1/2 hours.
Lay out a large sheet of parchment paper and brush with olive oil. Transfer the dough to the parchment and pat into a 6-by-12-inch rectangle. Brush the dough lightly with olive oil. Cover with plastic wrap and set aside 10 minutes. (At this point, you can refrigerate the dough, covered, up to 12 hours.)
Preheat the oven to 350 degrees F. Line 2 baking sheets with parchment; brush with olive oil. Cut the dough in half crosswise with a chef's knife or pizza cutter to make two 6-inch squares. Cut each half into 10 equal pieces.
On a lightly floured surface, stretch and roll each piece of dough with your hands into a thin 12-to-14-inch-long rope. Transfer the ropes to the prepared baking sheets, arranging them about 1/2 inch apart.
Whisk the egg white with 2 tablespoons water in a small bowl. Brush the breadsticks lightly with the egg wash, then sprinkle with parmesan. Cover loosely with plastic wrap; set aside 10 minutes.
Uncover the breadsticks and bake until golden and crisp, about 30 minutes, rotating the baking sheets halfway through. Slide off the baking sheets and let cool on racks. (The breadsticks will crisp up as they cool; if they are still flexible, bake about 7 more minutes.) Store in an airtight container up to 3 days.