In a small bowl, whisk together 3 tablespoons of olive oil with the mashed roasted garlic paste and a pinch of salt. Set aside — this is your flatbread base.
Heat a grill pan over medium-high heat. Toss the zucchini, peppers, corn, and red onion with the remaining olive oil, salt, and pepper.
Grill the vegetables in batches until charred and tender — about 3–4 minutes per side for the zucchini, peppers, and onion. The corn just needs a quick char, 2–3 minutes total. Set aside.
Brush both sides of the flatbreads lightly with plain olive oil and warm them on the grill pan for 1–2 minutes per side, until lightly toasted with grill marks.
Generously brush the tops of the warm flatbreads with the roasted garlic oil, making sure to spread it edge to edge.
Pile on the grilled vegetables — don't be shy.
Scatter the crumbled goat cheese generously over the top.
Finish with torn fresh basil, a sprinkle of chili flakes, and a generous drizzle of balsamic glaze.
Slice and serve immediately.