Whisk together white balsamic vinegar, minced garlic, brown sugar, salt, and freshly ground pepper in a large bowl until sugar is dissolved. Add olive oil in a slow, steady stream, whisking constantly until smooth. Add 1/3 cup chopped fresh basil and mozzarella cheese balls; toss to coat. Cover and chill 2 hours. Stir in strawberries; let stand at room temperature 30 minutes, stirring occasionally. Serve over fresh arugula or as an appetizer with toasted French bread baguette slices. Garnish with thinly sliced fresh basil.