Heat gas or charcoal grill. Brush 2 tablespoons of the oil over mushrooms; sprinkle evenly with salt and pepper. Brush remaining 2 tablespoons oil over bread slices. Set aside.
Cut bell pepper lengthwise into quarters; remove stem, seeds and membranes. Place pepper wedges, skin side down, on grill over medium heat. Cover grill; cook about 10 minutes or until skin is blackened. Place peppers in bowl; cover tightly with plastic wrap. Cool 5 minutes. Peel off blackened skin; rinse quickly with water. Cut pepper into thin slices; set aside.
Place mushrooms on grill over medium heat. Cover grill; cook 8 to 10 minutes, turning once, until tender. Add bread slices to grill for last 2 minutes of cooking time, turning once, until toasted. Cut mushrooms into 1/4-inch-thick slices. Spread goat cheese on 1 side of each bread slice. Top with mushroom and pepper slices. Sprinkle with basil. Drizzle with balsamic glaze.