Heat oven to 375°F and grease, or spray with nonstick spray, a muffin tin.
In a medium bowl, combine the Colby-Jack and Italian shredded cheeses. Toss gently to mix.
Unroll the bread dough on a lightly floured surface; press perforations to seal. Roll the dough to a 12x8-inch rectangle and cut it lengthwise in half.
Spread half of the cream cheese over each rectangle to within ¼-inch of edges.
Sprinkle each half with half of the shredded cheeses.
Starting with the long sides of each rectangle, roll the dough up like a jelly roll. Pinch the seams to seal and cut rolls into 1½-inch slices.
Place the lightly beaten egg and the shredded Parmesan in separate small bowls.
Dip the cut side of each slice in the egg and then in the Parmesan.
Place in the greased muffin cups, cheese side up.
Bake 18-22 minutes or until golden brown. Serve immediately.