Preheat a grill pan over medium heat. Preheat a sandwich/panini press as well if desired.
Slice eggplant on the bias (at a 45° angle) into 1/4-inch strips. Slice onion, pepper and squash into rounds. Peel carrots and cut into sticks.
Brush the sliced vegetables (all but carrots) with olive oil; season with salt and pepper then place on grill pan. Cook about 3 minutes per side until grill marks are present and they're slightly softened, then remove from heat and place on a baking sheet.
In a large bowl, whisk together minced garlic, Italian seasoning, and vinegar; drizzle over grilled vegetables.
Cut a 12-inch baguette into 4 equal pieces and slice open.
Spread pesto equally onto both halves of each bread slice.
Top bottom portions of bread with a single layer of each vegetable, then add cheese. Top with remaining bread slices.
Place sandwiches in a panini press and cook about 5 minutes each until well toasted (or use the grill pan, placing a heavy skillet on top of each sandwich to help press it down).
Serve with carrot sticks and frozen mango on the side.