Heat a large skillet over medium-high heat. Cook and stir beef in the hot skillet until browned and crumbly, 5 to 7 minutes; drain and discard grease. Let ground beef cool until easily handled.
Preheat oven to 350 degrees F (175 degrees C).
Process ham and American cheese in a food processor until finely chopped. Mix ground beef, ham and American cheese, pizza sauce, and oregano together in a large bowl. Spread mixture onto the bottom half of each hamburger bun and place the tops over the filling. Arrange hamburgers on a baking sheet.
Bake in the preheated oven until cheese is melted in the burgers and buns are lightly browned, 15 to 20 minutes.