In the bowl of a stand mixer fitted with the dough hook, whisk together the flour, yeast, salt, and onion powder. In a medium bowl, combine the water and oil.
Mixing at low speed, slowly add the oil-water liquid to the dry ingredients until a cohesive dough forms, about 2 minutes. Scrape down the bowl as needed, no dry flour should remain.
Increase the speed to medium-low, then knead until the dough becomes smooth and elastic, about 8 minutes.
Reduce the speed back to low, and slowly add 1 cup of the grated Parmesan in ¼ cup increments, until the Parmesan is mostly incorporated (about 2 minutes).
Transfer the dough to a clean, lightly floured work surface. Knead the dough by hand until the Parmesan is evenly distributed through the dough, which should form a smooth, round ball. This should only take 30 seconds. Lightly grease a large bowl or food-grade plastic container, place the dough ball in the bowl seam-side down, cover with a clean dish towel or plastic wrap and let rise for 1-1.5 hours, until doubled in size. [The dough can also be refrigerated for 8-16 hours at this point, then it needs to sit at room temperature for 1 hour before moving to the next step.]
Gently press down on the risen dough to deflate it. Transfer the dough to your clean work surface (not floured), and divide it first in half, then stretch each half into a 9-inch log and segment out 9 equal pieces (2 ounces each). Cover with greased plastic.
Work with one piece of dough at a time and keep the rest covered to avoid formation of a skin. Form the piece of dough into a rough ball shape by stretching the dough around so the edges meet along the bottom. Place the ball seam side down on the clean work surface, cup the ball with your hand, and drag it in tight counterclockwise circles until the dough feels taut. Repeat with all the pieces of dough, ensure they are all covered with the greased plastic, and let rest for 30 minutes.
Line two baking sheets with parchment paper.
Working with one dough ball at a time, stretch then roll it out using the palms of your hands into an 8-inch cylinder. Move your hands in opposite directions along the cylinder (one back, one forward) to twist the dough and form twisty breadsticks with a rounded point.
Adjust the oven racks to the upper-middle and lower-middle positions. Start pre-heating the oven to 500 F.
Arrange the breadsticks 1 ½ inches apart on the prepared baking sheets. Cover them with the greased plastic and let them use until almost double in size, about 30 minutes. The dough should spring back only minimally when poked gently.
Whisk together the egg, olive oil, and salt in a small bowl. Gently brush the risen breadsticks with the egg wash using a pastry brush. Sprinkle them with the remaining ½ cup of grated Parmesan.
Bake in the pre-heated oven until golden brown, about 12-14 minutes total. Switch and rotate the sheets halfway into baking. Transfer the baked breadsticks to a wire rack and let them cool about 10 minutes. Serve warm.
These Parmesan breadsticks will kept in a sealed Ziploc bag for 3 days at room temperature. Alternatively, they can be wrapped in aluminum foil then frozen for one month. To reheat (from thawed if previously frozen), baking in a 350 F oven for 10 minutes.