Brush mushrooms, onion, and squash with oil; sprinkle with salt and pepper. For a charcoal grill, grill vegetables directly over medium heat, turning once. Allow 10 to 12 minutes for mushrooms and onion and 5 to 6 minutes for squash. (For a gas grill, preheat grill. Reduce heat to medium. Place vegetables on grill rack over heat. Cover and grill as above.) Meanwhile, mash avocado in a small bowl; stir in pesto.
Spread avocado-pesto on cut surfaces of top and bottom slices of ciabatta. Thickly slice mushrooms. To assemble, on bottom bread slice layer half the mozzarella, mushrooms, onions, squash, center bread slice, remaining mozzarella, roasted red peppers, spinach, then ciabatta top. Press together and wrap in foil. Heat directly over medium coals for 15 minutes, turning twice, until heated through. Slice to serve. Makes 4 to 6 servings .