1. Unwrap burrata; gently pat dry. Wrap burrata in several layers of plastic wrap to preserve the "ball" shape. Freeze 8 hours or until completely frozen.
2. Preheat oven to 425°.
3. Combine 2 teaspoons oil and next 6 ingredients (through garlic); toss to combine. Arrange tomatoes, cut sides up, on a foil-lined baking sheet. Bake at 425° for 40 minutes or until blistered and liquid almost evaporates. Cool. Chop tomato mixture.
4. Reduce oven temperature to 350°.
5. Working with 1 phyllo sheet at a time (cover remaining phyllo to prevent it from drying), lay 1 sheet on a flat surface, and coat with cooking spray. Stack another phyllo sheet at a slight angle over the first, and coat with cooking spray. Repeat the procedure with remaining phyllo and cooking spray, angling each phyllo sheet slightly over the previous sheet. Using a slotted spoon, spoon tomato mixture in center of phyllo stack, and discard any liquid. Unwrap burrata, and place frozen burrata on tomato mixture. Fold the phyllo edges over to enclose cheese and tomato mixture, pressing to seal. Turn ball over with tomato mixture on top, and place on a foil-lined baking sheet coated with cooking spray. Brush phyllo with remaining 1 tablespoon olive oil. Bake at 350° for 30 minutes. Remove from oven.
6. Preheat broiler to high.
7. Broil cheese ball 2 minutes or until golden brown. Serve immediately.