In a small bowl, whisk together the olive oil, garlic, and Italian seasoning.
On a lightly floured surface, roll out the pizza dough into a 12-inch square. Brush the entire surface with the seasoned olive oil. Top with 1 cup of shredded cheese and an even layer of ham. (Reserve the extra cheese and ham for Round 2 Ham and Cheese Croquettes.)
Starting from the bottom edge, tightly roll up the dough to form a log. Using a sharp serrated knife, slice the roll on the bias into 1-inch thick pieces. Arrange the rolls on their side on a nonstick or lightly oiled baking sheet. Cover loosely with plastic wrap and allow to sit at room temperature for 30 minutes to proof.
Preheat the oven to 375 degrees F.
In a small bowl, whisk together the egg with 2 tablespoons of water. Brush the tops and sides of the rolls with the egg wash. Bake until golden brown, about 30 to 35 minutes.
Remove from the oven, transfer to a serving platter and serve.