Combine the flour, yeast, salt and sugar in the bowl of a food processor and pulse once or twice. Add 2 tablespoons oil and pulse a few more times. With the machine running, add 1 cup water through the feed tube. Continue to add water, a tablespoon at a time, until the mixture forms a ball. The dough should be a little rough and rather sticky.
Put some olive oil in a bowl and transfer the dough ball to it, turning to coat well. Cover with plastic wrap and let it rise in a warm place until about double in size, about 1 hour. Knead lightly, forcing out the air, for about 3 minutes. Reshape the dough into a ball, return it to the bowl, cover and let rise in the refrigerator for several hours or up to overnight.
Heat the oven to 400 degrees F. Lightly grease two baking sheets with olive oil and sprinkle lightly with semolina.
Cut the dough into 3 pieces; keep the remaining 2 pieces covered while you work with the first. On a well-floured surface, roll a piece of dough out as thinly as possible into a large rectangle about 8 inches long. Use a sharp knife or pastry wheel to cut the dough into roughly 1/4-inch wide breadsticks (slightly smaller is better than slightly bigger).
To roll thinly with a pasta machine, roll out the dough to a 1/4 inch thickness by hand. Put the dough through the machine with the rollers fixed on the largest setting, then cut the dough into breadsticks roughly 1/4 inch wide and 12 inches long. Repeat with the remaining dough.
Once you've cut each strip, twist it, as though you were wringing out a washcloth, to create twisted rods. Transfer to the baking sheets, spaced apart, and brush with olive oil. Bake until crisp and golden, 10 to 20 minutes. Cool completely and serve. Breadsticks will keep in an airtight container for up to 1 week.