Preheat the oven to 400°F (200°C). Line a large baking sheet with parchment paper.
Roll each tube of pizza dough into a rectangle about 10 x 14 inches. Cut each rectangle into 4 equal rectangles for a total of 8 pieces.
Spread about 1/2 teaspoon Dijon mustard on one half of each rectangle, leaving a 1/2-inch border around the edges.
Divide the ham evenly among the rectangles.
Top each portion with 1 slice of provolone cheese and about 2 tablespoons shredded cheddar cheese.
Sprinkle the garlic powder, onion powder, and black pepper evenly over the filling.
Fold the dough over the filling and press the edges together firmly. Crimp with a fork to seal.
In a small bowl, whisk together the egg and water. Brush the tops of the hot pockets with the egg wash.
Cut two small vents in the top of each pocket.
Bake for 18 to 22 minutes, or until puffed and deep golden brown.
Allow the hot pockets to cool for 5 minutes before serving.