Whisk The Balsamic Vinegar, Oil, Salt, And Pepper In A Small Bowl. Pour Over Tomato Slices To Marinate. Preheat An Outdoor Grill For Medium-High Heat, And Lightly Oil The Grate. Mix Ground Beef, Tomato Paste, Basil, Parmesan Cheese, Garlic, And 1/4 Teaspoon Pepper In A Large Bowl. Form Beef Mixture Into 4 Equal Patties. Cook On The Preheated Grill Until The Burgers Are Cooked To Your Desired Degree Of Doneness, About 5 Minutes Per Side For Well Done. An Instant-Read Thermometer Inserted Into The Center Should Read 160 Degrees F (70 Degrees C). Top Each Burger With Mozzarella Cheese; Allow To Melt. Serve On Hamburger Buns With Marinated Tomato Slices.