Place 1 tablespoon parsley, shallots, salt, and garlic in a mortar or small bowl; grind with a pestle until mixture forms a smooth paste. Stir in piquillo peppers and tomato. Add vinegar, oil, mustard, and black pepper; stir well to combine. Place asparagus in a shallow dish, and drizzle with pepper mixture. Gently toss asparagus and pepper mixture, using a serving fork and spoon. Place asparagus on a serving platter.
Cut egg in half lengthwise; remove and reserve the yolk for another use. Chop egg white. Sprinkle egg white, capers, and remaining tablespoon of parsley evenly over asparagus.