Preheat oven to 375°. In a small bowl, beat cream cheese, garlic powder and onion powder until blended. In another bowl, toss Colby-Monterey Jack cheese with Italian cheese blend.
On a lightly floured surface, unroll breadstick dough; press perforations to seal. Roll dough to a 12x8-in. rectangle; cut dough lengthwise in half.
Spread each 12x4-in. rectangle with half of the cream cheese mixture to within 1/4-in. of edges. Sprinkle each with half of the combined cheeses; roll up jelly-roll style, starting with a long side. Pinch seam to seal. Cut rolls into 1-in. slices.
Place beaten egg and Parmesan cheese in separate shallow bowls. Dip a cut side of each slice in egg, then in Parmesan cheese; place slices in greased mini muffin cups, cheese side up.
Bake until golden brown, 17-20 minutes. Serve warm.