Even the Italians love hamburgers. But in the Italian culture it cannot be only about meat. The Italian meal needs to have balance, so the addition of a little vegetable and a little cheese rounds out this meaty meal. For the record, the Italian-Mediterranean diet is the ideal diet because of its diversity and moderation.
In a large bowl, combine the ground beef, sun-dried tomatoes, vinegar, and salt. Mix the ingredients together with your hands and form into 6 equal-sized balls.
In a small bowl, toss together the mozzarella and basil. Make an indentation in each ball and press in some of the mozzarella-basil mixture. Seal the mozzarella in and press to form patties about 1-inch thick.
Heat a large cast-iron skillet over medium-high heat. Add the olive oil. When the oil is hot, add the patties and cook until cooked through and the cheese inside is melted, about 8-10 minutes. To check if the cheese has melted inside, stick a toothpick into the center; if the cheese has melted it will stick to the toothpick. Serve with the homemade ketchup on a bun or with a salad.