Preheat oven to 450°F. Line a 15 x 10 x 1-inch baking pan with enough parchment paper to overhang the sides.
Combine 1-1/2 cups flour, undissolved yeast and salt in a large bowl. Heat water, milk, oil and corn syrup until very warm (120° to 130°F). Add to flour mixture; mix until well blended, about 1 to 2 minutes. Gradually add enough remaining flour until dough forms a ball. Knead on lightly floured surface, adding additional flour if necessary, until smooth and elastic, about 3 to 4 minutes.
Divide dough in half. Roll one portion to 15 x 10-inch rectangle to fit prepared baking pan. Place dough on pan, stretching as needed to meet edges. Sprinkle with 3 cups cheese, leaving a 1-inch border. Roll out remaining dough to slightly larger rectangle than the first. Place over the cheese and fold the edges under the bottom layer of dough; pinch to seal. Cut crosswise through the dough with a sharp knife to create 20 breadsticks, but do not separate the breadsticks. Cover and let rise 20 minutes.
Whisk egg whites with oregano and garlic; brush over dough. Sprinkle evenly with remaining cheese. Bake 15 to 17 minutes until puffed and golden. Cut along seams into breadsticks; serve warm.
TIP: Serve with warm marinara sauce for a special appetizer.