Grill the Sourdough: Lightly brush each side of the sourdough slices with olive oil. Place them on a hot grill pan over medium-high heat. Cook for 2-3 minutes per side, until golden brown with distinct grill marks. Set aside on a wire rack to cool slightly.
Prepare Fresh Toppings: While the bread grills, prepare the other components. Halve the cherry tomatoes and gently tear the mozzarella and fresh basil leaves into rustic, bite-sized pieces. Arrange them in separate small bowls for easy assembly.
Whip the Avocado Base: Halve the avocados, discard the pits, and scoop the flesh into a medium bowl. Add the fresh lime juice, fine sea salt, black pepper, and optional red pepper flakes. Vigorously mash with a fork until it reaches a creamy, slightly textured consistency.
Build the Base Layer: Divide the whipped avocado mixture evenly among the 6 slices of grilled sourdough. Spread it from crust to crust, creating a generous foundation for the toppings.
Arrange the Caprese Elements: Artfully distribute the torn mozzarella pieces and halved cherry tomatoes over the avocado layer on each toast.
Garnish and Serve: Finish by scattering the torn basil leaves across the toasts. Apply a final, delicate drizzle of high-quality balsamic glaze over everything. For the best texture and flavor, serve immediately.