Make the dough: Whisk the sugar and 3/4 cup warm water (about 105 degrees F) in a bowl. Sprinkle the yeast on top and set aside until foamy, about 10 minutes. Stir in the vegetable oil. Whisk the flour and salt in a large bowl. Make a well in the center, pour in the yeast mixture and stir with a wooden spoon to make a shaggy dough. Turn out onto a generously floured surface and knead until smooth and elastic, about 5 minutes, dusting with more flour as needed. Form into a ball. Brush a large bowl with vegetable oil; add the dough, turning to coat with the oil. Cover tightly with plastic wrap and set aside at room temperature until doubled in size, about 1 hour, 30 minutes.
Make the filling and topping: Toss 1/2 cup mozzarella, the cheddar, pecorino, garlic salt, granulated garlic and dried parsley in a bowl. Preheat the oven to 450 degrees F. Put an inverted baking sheet in the oven; preheat for 10 minutes. Line another inverted baking sheet with parchment paper and sprinkle with the cornmeal. Cut the dough in half; transfer 1 piece to the parchment-lined baking sheet and stretch into an 8-inch square (keep the other piece of dough covered). Sprinkle 1 side of the square with 1/2 cup of the remaining mozzarella, then fold the dough in half to enclose the cheese, pressing to seal. Slice the dough crosswise at 1-inch intervals, stopping just before the folded edge. Sprinkle with half of the cheese-garlic mixture. Move the dough to 1 side of the baking sheet. Repeat with the remaining dough, mozzarella and cheese-garlic mixture on the other side of the baking sheet. Slide the bread on the parchment onto the hot baking sheet in the oven. Bake until the dough is cooked through and the cheese is golden brown, 12 to 14 minutes. Let cool slightly, then pull apart into breadsticks.