Heat the oven to 425°F with a rack in the center. Line a large light-colored sheet pan with parchment.
Blend cottage cheese and olive oil until completely smooth, scraping the container as needed.
Whisk 1 1/2 cups flour, baking powder, salt and 1/2 teaspoon garlic powder. Stir in blended cottage cheese until a shaggy soft dough forms.
Rest dough 5 minutes. If it coats a floured finger and slumps, fold in additional flour 1 tablespoon at a time, stopping once it holds shape.
On a lightly floured surface, fold dough 5 or 6 times. Divide into 8 portions and roll each into a 7-inch rope. Space ropes 1 1/2 inches apart.
Beat egg with water and brush a thin coat over the tops and sides without puddling at the base.
Bake 14 to 17 minutes, rotating after about 9 minutes, until golden and no longer gummy inside; the thickest center should be about 200°F.
Mix melted butter with remaining garlic powder. Brush hot breadsticks, sprinkle with Parmesan and optional parsley, and cool 5 minutes.
