Dissolve 1 1/2 teaspoons sugar and yeast in 1/2 cup warm water in a bowl. Let stand until yeast softens and begins to form a creamy foam, about 5 minutes.
Beat flour, oil, egg, salt, remaining sugar, and remaining warm water into yeast mixture until smooth, adding more flour if needed to form a soft dough. Turn dough onto a floured work surface; knead until smooth and elastic, 6 to 8 minutes.
Place dough in a greased bowl, turning once to cover completely with oil. Cover bowl with a damp towel and place in a warm area until dough is doubled in size, about 40 minutes.
Grease a baking sheet.
Punch dough down and turn onto a floured work surface. Divide dough into 18 pieces and shape each into a 6-inch rope. Arrange dough pieces on the prepared baking sheet about 2 inches apart; cover with a towel or plastic wrap and let rise until doubled in size, about 25 minutes.
Preheat oven to 400 degrees F (200 degrees C). Remove cover from dough.
Bake in the preheated oven until golden brown, 10 to 12 minutes.
Whisk butter and garlic salt together in a bowl; brush over each breadstick. Sprinkle Parmesan cheese over each.