Preheat your grill or grill pan over medium-high heat.
In a large bowl, combine the zucchini, yellow squash, red bell pepper, and red onion. Drizzle with olive oil and season with salt and pepper. Toss to coat the vegetables evenly.
Place the seasoned vegetables on the grill. Cook for about 5-7 minutes, turning occasionally, until they are tender and have nice grill marks. Remove from the grill and set aside.
While the vegetables are grilling, toast the bread slices on the grill for about 1-2 minutes on each side until golden brown.
Assemble the sandwiches by layering the grilled vegetables on two slices of toasted bread. Top with slices of fresh mozzarella cheese.
Drizzle balsamic glaze over the cheese and add fresh basil leaves. Top with the remaining slices of toasted bread.
If desired, return the assembled sandwiches to the grill for an additional 2-3 minutes to melt the cheese slightly.
Slice the sandwiches in half and serve warm.