In the bowl of a stand mixer, fitted with the hook attachment, add the sugar and water. Sprinkle the yeast on top of the water and let stand for 5-10 minutes until the yeast blooms (it should appear dissolved and foamy around the edges).
Add the flour, egg, salt, and butter to the bowl of the stand mixer, and mix on low. Scrape down the sides of the stand mixer. Continue to knead the bread on low for 8-10 minutes. After kneading, the dough should no longer stick to the sides of the mixing bowl, and it should feel soft but not sticky.
Form the dough into a smooth ball, and place the dough in large bowl sprayed with cooking spray. Cover the dough with a clean kitchen towel. Allow the dough to rise for one hour, or until about double in size.
On a floured surface, roll dough into a 12" x 16" inch rectangle. Using a bench scraper, slice the dough in half lengthwise. Then, make eight equal cuts cross-wise to make 16 equal rectangles of dough.
Using your fingers, pinch the long edges of each piece of dough together to form a rounded breadstick. Place each breadstick, seam side down, on a half-sized baking sheet sprayed with cooking spray. Cover the breadsticks with the kitchen towel and allow to rise for another 30 minutes.
Preheat the oven to 350 degrees. Bake the breadsticks for 15-18 minutes until they are cooked through and golden brown on the top.
Using a pastry brush, brush the melted butter on top of the breadsticks. Sprinkle the garlic salt and Italian seasoning on top of the breadsticks, adding more to taste.