Grilling imparts a robust, nutty flavor to avocado. Choose a firm, ripe one.
Brush 1/2 avocado (pitted and peeled) with fresh lime juice and olive oil. Grill, cut side down, for 1 to 2 minutes, creating grill marks. Turn to create crosshatch marks, and grill for 1 minute more. Meanwhile, brush 1 side of a slice of bread with olive oil, and grill, about 30 seconds per side. Top bread with thin slices of parmesan and the grilled avocado. Mash with a fork, brush with olive oil and salt.