In a large bowl, add the ground chuck, onion and oregano. Sprinkle the salt evenly over the bowl and add a hefty amount of black pepper. Gently mix, divide into 4 balls and allow to dry-marinate at room temperature for 1 hour.
Preheat a gas grill or large griddle on high.
Divide each ball in half to create 8 smaller balls, then press each into a thin flat patty between 1/4 and 1/8 inch thick. The easiest way to do this is between 2 pieces of parchment paper, using a glass or rolling pin. Brush the top side of each patty with the olive oil. Add the patties oiled-side down to the grill and cook until the edges are golden, about 5 minutes. Top half of the patties with the Parmesan and cover each with another patty, creating 4 sandwiched patties. Turn off the heat and cover with aluminum foil or an upside-down metal bowl to tent for just 2 minutes.
Place the patties on a garlic bread round (crust-side down) and top each with some tomatoes and basil and a drizzle of balsamic syrup. Top with another garlic bread round (crust-side up) and serve.
Make a simple, savory garlic bread pudding with the leftover scraps of garlic bread, 1 1/2 cups heavy cream and 8 ounces scallion-flavored whipped cream cheese. Mix in a bowl and divide the mixture among 4 buttered ramekins. Bake in an oven preheated to 350 degrees F until bubbly on the edges and set in the center, 25 to 30 minutes. Or make croutons for tomato soup or Caesar salad by cutting the scraps into small cubes and toasting them in a dry pan.