Preheat a grill pan or outdoor grill to medium-high heat (about 400°F/200°C).
In a bowl, toss the sliced zucchini, red bell pepper, and red onion with 1 tablespoon of olive oil, Italian seasoning, salt, and black pepper until evenly coated.
Grill the vegetables for 3-4 minutes per side, or until tender and slightly charred. Remove from the grill and set aside.
Brush one side of each bread slice with the remaining olive oil. If using pesto, spread a thin layer on the unoiled side of two slices.
Assemble the paninis: On the pesto side (or unoiled side) of two slices, layer the grilled vegetables and sprinkle with shredded mozzarella cheese. Top with the remaining bread slices, oiled side up.
Heat a panini press or a skillet over medium heat. Place the sandwiches in the press or skillet. If using a skillet, press down with a spatula and cook for 3-4 minutes per side, until the bread is golden and the cheese is melted.
Remove from the heat, cut each panini in half, and serve warm. Enjoy your grilled vegetable panini!