In the bowl of a stand mixer, combine warm water, yeast and sugar. Let stand for 5 minutes. Stir in olive oil and salt.
Gradually add the flour until a soft dough forms. Let dough raise for 10 minutes.
Meanwhile, stir together melted butter and minced garlic.
On a lightly floured surface, roll dough into a large square, about 16×16 inches. Brush dough with half of the garlic butter. Sprinkle with Parmesan cheese.
Fold dough in half, then cut into 1 inch wide strips. Twist strips and place on 2 lightly greased baking sheets. Allow breadsticks to raise for 15-20 minutes while you preheat the oven to 400 degrees.
Bake for about 18-20 minutes, or until golden brown. Brush breadsticks with remaining garlic butter and sprinkle with coarse salt.