Trim the asparagus and peel if necessary. Cook in boiling salted water for 3 minutes or until just tender but still quite firm. Drain and fresh in iced water. When completely cold, drain the asparagus and pat dry on a clean cloth. Set aside. Add a dash of vinegar to a small pan of simmering salted water, then poach the eggs for the sauce. Drain and refresh in cold water. Drain on paper towels and leave to cool. Lightly heat a non-stick frying pan that is about 20cm in diameter. Brush with a film of olive oil then sprinkle in a thin, even layer of Parmesan. Cook for about 2 minutes or until the cheese melts and turns golden. Use a palette knife (metal spatula) to transfer the Parmesan cracknel to paper towels. Repeat to make 8 cracknel in all. Prepare a charcoal grill. To finish the sauce, put the poached eggs in a bowl with the remaining ingredients and crush finely with a fork. Season with salt and pepper. Toss the asparagus in olive oil and season with salt and pepper. Chargrill to mark them on all sides. Cut the beef into very thin slices. Divide among 4 flat plates, arranging them in one layer if possible and cover with clingfilm (plastic wrap). Pound until the beef is very thin. Remove the film and brush the beef with olive oil. Sprinkle with lemon juice and season with sea salt and pepper. Place a Parmesan cracknel on top of each carpaccio, then add the grilled asparagus and sprinkle with basil. Top with the remaining cracknel. Spoon the egg sauce around and serve.