Dissolve sugar and yeast in 1 cup plus 1 tablespoon warm water in a bowl. Let stand until yeast softens and begins to form a creamy foam, about 5 minutes.

Combine flour and salt in the bowl of a stand mixer fitted with the paddle attachment. Mix 2 tablespoons butter into flour mixture until incorporated. Pour yeast mixture into flour mixture and knead dough using the dough hook attachment until smooth and elastic, about 10 minutes.

Place dough in a container and cover with a towel or plastic wrap; let stand until doubled in size, 1 to 1 1/2 hours.

Line a baking sheet with parchment paper.

Divide dough into 2-ounce portions and roll into 7-inch long sticks. Arrange sticks on the prepared baking sheet; cover with plastic wrap and let stand until dough doubles in size, 1 to 1 1/2 hours.

Preheat the oven to 400 degrees F (200 degrees C). Remove plastic wrap from dough.

Bake in the preheated oven until breadsticks are golden brown, 10 to 12 minutes. Brush each breadstick using remaining 2 tablespoons melted butter; top with garlic salt.