Prepare a grill for medium-high heat. It's best to use a grill basket, but if you don't have one, oil the grill grates.
Cut the tomatoes in half crosswise and brush with 4 tablespoons of the olive oil. Place the tomatoes cut-side down on the grill (or grill basket) and cook until the tomatoes are softened and charred, about 4 minutes. Flip once and continue to cook for 1 to 2 minutes more. The tomatoes should be tender.
Place the tomatoes in a serving bowl cut-side up and season generously with salt and pepper. Place the burrata in the center of the tomatoes and cut in half. Garnish with the basil and oregano. Sprinkle with lemon zest and drizzle with the balsamic glaze and remaining 2 tablespoons olive oil. Serve with crusty bread.