To make the Parmesan butter, tip the lemon juice into a small saucepan and, over a very low heat, add the butter. Melt and stir the butter into the lemon juice until you have a smooth sauce, add the Parmesan and season to taste, then set aside.
To make the Asian dressing, using a pestle and mortar, crush the chili with the brown sugar, then stir in the lime juice until the sugar has dissolved. Stir through the soy sauce and sesame oil, then set aside.
For the garlic mayonnaise, mix all the ingredients together. Decant the sauces into 3 serving dishes.
Bring a large pan of water to the boil and cook the asparagus for 2-3 mins until just tender. Drain well and serve steaming hot in a pile for everyone to help themselves and dip into the sauce of their choice.