Add vinegar and brown sugar, if using, to a small saucepan and bring to a boil.
Reduce to a very low simmer and cook for 10-15 minutes, until liquid reduces by about half and is slightly syrupy. Remove from heat, pour vinegar in a bowl or glass to be used later, and set aside to cool and thicken.
Preheat oven to 450 degrees F (230 degrees C) or grill to high heat.
Place the slices of bread on a baking sheet and drizzle with olive oil on both sides. Rub each side of bread with the halved garlic clove. Sprinkle with salt and pepper to taste.
Drizzle avocado with olive oil, salt and pepper to taste.
Place bread slices and avocado slices in oven or grill both until lightly charred, about 2 to 3 minutes per side. Remove from the oven or grill.
Slice Stella fresh mozzarella and divide equally among the sandwiches. Place on top toast, then add fresh basil to taste. Add grilled avocado slices and tomato slices. Drizzle everything with olive oil, balsamic drizzle and salt and pepper to taste.