Combine the yeast, along with ¾ cup of the warm water, sugar, and ½ cup of the flour, in the bowl of a stand mixer (or large mixing bowl if preparing by hand). Mix well and set aside to proof for about 10 minutes.
When the yeast is frothy, add the remaining water and flour, Parmesan, Italian seasoning, and salt. Beat at low speed using the mixer’s paddle attachment (or with a wooden spoon) until the flour is incorporated and a soft dough forms.
Replace the mixer’s paddle attachment with the dough hook and knead at medium speed for about 2 minutes. Increase the speed to high and continue kneading for an additional 3 to 4 minutes or until the dough forms a soft ball and is springy (if kneading by hand it will take about 10-12 minutes total). Check the dough several times during machine kneading and if it seems too soft, add a tablespoon or two of flour.
Lightly grease a large mixing bowl with olive oil. Add the dough, turning it so that all sides are coated with oil. Cover the bowl with plastic wrap and set aside to rise for 1 to 2 hours or until doubled.
Generously oil a lipped baking sheet. Punch the dough down and gently press it into the oiled baking sheet. If the dough resists being pressed into the baking sheet, allow it to rest for 5-10 minutes until relaxed enough to work with. Using a pizza wheel or very sharp knife, cut the dough lengthwise into about 1” strips. Make one final cut crosswise. Cover the pan and let the breadsticks rise for about 30 minutes.
When ready to bake, preheat the oven to 400 degrees. Beat the egg and water together and brush over the top of the breadsticks. Sprinkle the dough with the pumpkin seeds, salt, and black pepper. Very gently press the pumpkin seeds onto the dough to help them adhere.
Bake for 25-35 minutes or until a deep golden color. Remove from the oven and cool for 5 minutes. Cut along the score lines with a sharp knife to separate the breadsticks.