In the bowl of a stand mixer fitted with a dough hook (or in a large bowl if mixing and kneading by hand), combine 1 ½ cups of the flour, yeast, salt, and sugar.
Heat the water and olive oil in a small saucepan until steaming. Remove from heat and let sit for 5 minutes, or until temperature is between 120-130°F. Slowly add the water and oil to the dry ingredients, while mixing on low speed. Use a spatula to scrape down the bowl as needed.
With the mixer on medium low, add the remaining flour a ¼ cup at a time, mixing until incorporated between each addition. Continue adding flour until the dough clears the bowl and is sticky to the touch without coming off in your hands.
Knead with a dough hook or by hand until smooth and elastic, about 5-8 minutes more.
Shape dough into a ball and place in a greased bowl and cover with plastic wrap. Put in a warm place and let rise until doubled, about 1 hour.
Gently push down on the center of the dough to release the gases and knead by hand on a clean surface a few times.
Divide dough into 12 pieces and roll each into a rope about 8 inches long. Place on a half sheet pan or inside a 9x13-inch pan that has been greased or lined with parchment paper. Cover with plastic wrap and let rise until doubled, about 30-45 minutes.
Preheat oven to 375°F. Brush breadsticks with pesto and sprinkle with parmesan cheese. Bake until golden, about 15 minutes. Let cool briefly before serving with additional pesto, if desired.
Note: These breadsticks are best the day they are made, but may be stored at room temperature for up to 3 days.