For the sauce: To a small saucepan over low heat add the oil and garlic and cook until fragrant. Add the tomato paste and seasoning, and mash with the back of a wooden spoon. Stir in the crushed tomatoes and bring to a simmer. Simmer for 15 minutes, then remove from the heat.
For the burger: Preheat grill to medium-high heat. Preheat oven to 550 degrees F with pizza stone on the middle rack of the oven.
Form the meat into four 4-ounce patties and sprinkle both sides with salt and pepper. Grill until medium, then remove from the heat and place on a paper-towel-lined plate.
Divide the pizza dough into four 4-ounce balls. Press out each ball on a floured pizza peel to about a 5-inch round. Top each with a slice of scamorza, a couple slices of pepperoni, then 1 burger patty and 1 tablespoon of pizza sauce. Fold the dough over the burger as if making a large dumpling. Brush with olive oil. Bake on the pizza stone in the oven until golden brown, 8 to 10 minutes. Serve with extra tomato sauce on the side to dip.