Prepare outdoor grill with medium-hot coals, or heat gas grill to medium-high, or see Stovetop and Oven Method, below. Lightly brush grill grid with vegetable oil.
In large bowl, gently toss together eggplant, squash, red onion, mushrooms, red pepper, oil, salt and pepper.
Grill eggplant, squash and onions, uncovered, for 3 to 4 minutes per side until fork-tender, turning a few times if necessary to avoid burning. Transfer to a large clean bowl; cover to keep warm.
Place whole mushrooms and red pepper in grilling basket. Grill until fork-tender, 6 to 8 minutes, turning a few times to avoid burning. Transfer to bowl with other vegetables.
In small bowl, whisk together vinegar, garlic, salt, pepper, mustard and brown sugar. Gradually drizzle in olive oil, whisking until thick and well blended. Add to vegetables; toss. Keep warm.
Toast bread 2 to 3 minutes per side. Rub with peeled garlic.
Divide the dressed vegetables among toasted bread slices. Top with the cheese.
Place open-faced sandwiches on grill over indirect heat. Heat through, about 5 minutes. Serve warm. Makes 6 servings.
Heat oven to 350 degrees F. Heat stovetop grill pan over medium-high heat. Brush pan lightly with vegetable oil. In batches, grill vegetables and bread as directed above in steps 3, 4 and 6. Place completed sandwiches on baking sheet in warm oven to heat through, about 3 minutes.