1. For dressing: Toast pimenton in a small skillet over medium heat, stirring frequently, until fragrant, about 4 minutes. Puree pimenton, piquillo peppers, mayonnaise, vinegar, honey, 1 teaspoon salt, and pepper, to taste, in a blender until smooth. Transfer mixture to a bowl, cover, and refrigerate.
2. For vegetables: Preheat grill or stovetop grill pan to medium-high. Brush both sides of the mushrooms with some olive oil and season with salt and pepper. Grill mushrooms, turning once, until tender and slightly charred, about 5 minutes. Brush scallions with olive oil, season with salt and pepper, to taste, and grill, turning occasionally, until soft and slightly charred, about 8 minutes. Transfer to a cutting board and roughly chop.
3. Grill bread on both sides until lightly toasted. Rub garlic and spread dressing onto one side of each piece toast. Layer scallions, mushrooms, and cheese on 4 toasts, and top with remaining bread. Press down slightly and cut in half; serve.
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