Step 1

Prepare grill for direct- and indirect-heat cooking. In a bowl, whisk together vinegar, mustard, and oregano; season with salt and pepper. Slowly whisk in oil; set aside.

Step 2

Brush mushrooms, zucchini, kale, and onion with more oil; season with salt. Grill mushrooms over direct heat 6 to 10 minutes, depending on size. Flip mushrooms; add zucchini and onion to grill and cook 5 minutes more. Flip zucchini and onion; add kale to grill.

Step 3

Continue cooking, flipping kale once, until everything is charred in places and cooked through, 3 to 4 minutes more. Transfer to a platter or board. Top flatbreads evenly with about two-thirds of mozzarella. Divide mushrooms and vegetables evenly among flatbreads. Sprinkle with remaining mozzarella and blue cheese.

Step 4

Place pizzas over indirect heat and grill, covered, maintaining a temperature of about 400°F, just until bread is charred in places and mozzarella melts, 7 to 9 minutes (if bread isn't crisping along bottom and edges, transfer pizzas to direct heat in last minute of cooking). Drizzle with vinaigrette and serve warm.