Preheat a grill pan to medium-high or a broiler to high. In a small saucepan, heat the EVOO over medium heat. Add the garlic and rosemary and let steep for 3 minutes.
In a large skillet, combine 1 tablespoon of the infused EVOO and the red onion over medium heat and cook until softened, 6 to 7 minutes. Add the tomatoes, brown sugar and vinegar, lower the heat and cook until thickened, 7 minutes.
Brush half the eggplant with the infused EVOO and season with salt and pepper. Grill or broil, flipping once, until tender, 2 to 3 minutes on each side. Repeat with the remaining eggplant.
Dress the arugula and basil with the lemon juice and remaining infused EVOO; season with salt and pepper.
On each roll bottom, layer 4 eggplant slices, a handful of greens and 2 smoked mozzarella rounds. Grill or broil to melt the cheese. Slather the roll tops with the tomato jam and set in place.