Heat butter in a skillet over medium heat. Add garlic and onion and sauté until soft and fragrant. Add chopped artichoke hearts and oregano and cook for another couple of minutes. Remove from heat and set aside.
Add grated caciocavallo, ricotta, whisked egg, salt and pepper to a bowl and mix to combine thoroughly.
Cut pizza dough into eight pieces and roll each piece into a six millimeter thick circle. Add 1/4 to 1/3 cup (60-80 milliliters) of filling to centre of circle, then fold dough over to create a crescent shape. Seal edges together firmly by rolling and pressing with your fingers or crimping with a fork.
Heat vegetable oil in a large saucepan to 375 F (190 C). Fry panzerotti in batches, turning periodically, until golden brown. Transfer to paper-towel lined platter. Allow to cool for a few minutes before serving.