Slice eggplant into 1/4 or smaller slices. Sprinkle with salt and let bitter juices draw out for about 10 minutes. Rinse and pat dry.
Heat the grill to 400. Brush the sliced eggplant and zucchini with olive oil and grill on both sides for about 1 minute each.
In a small mixing bowl, mix together the mayonnaise, lemon juice, olive oil, crushed garlic, rosemary and a pinch of salt and pepper.
Assemble panini sandwiches by spreading aioli onto the insides of the bread, then adding the grilled eggplant, grilled zucchini, a slice of cheese. Brush the outsides of the bread with olive oil.
Grill the paninis for 2-3 minutes per side, or until golden brown and cheese has melted.