Pour balsamic vinegar into a small saucepan and bring to a boil over medium heat.
Reduce heat to medium-low, remove lid, and simmer uncovered for 10-12 minutes, stirring occasionally, until reduced by half and coats the back of a spoon.
Pour into a bowl to cool.
Rinse Roma tomatoes, halve lengthwise, scoop out pulp and seeds, and finely dice the flesh.
Mix diced tomatoes with minced garlic, salt, pepper, and white balsamic vinegar.
Stack basil leaves, cut into ribbons, and gently fold into tomato mixture.
Let tomato mixture stand at room temperature.
Position oven rack 6 inches from broiler and turn broiler on high.
Slice baguette diagonally and lay slices on a baking sheet.
Brush both sides of bread with olive oil.
Broil both sides until lightly golden, watching closely.
While bread is warm, rub both sides with peeled garlic cloves.
Turn off broiler and preheat oven to 400F.
Place one slice of mozzarella on each baguette slice.
Bake for 4-5 minutes until cheese is melted.
Remove from oven and sprinkle lightly with salt.
Top each piece with a spoonful of tomato mixture and drizzle with balsamic reduction.
Serve immediately while bread is crispy.