First, make the mayonnaise. In a food processor, add the whole olives, egg yolks, vinegar, mustard and anchovy, and blend until you have a smooth paste. Slowly add the oil until it becomes thick and glossy. To finish, stir through the chopped olives, lemon juice and cayenne pepper.
Next, in a pan over a low heat, warm the clarified butter until it melts. Put the flour, beaten eggs and brioche crumbs into three separate bowls. Dip the asparagus stem into the flour, leaving the tip uncovered, then dip it into the beaten egg and finally into the brioche crumbs. Repeat for each spear.
Turn up the heat until the melted butter is hot. Add four spears of asparagus and fry for 1 min or so – it will cook quickly. Use a slotted spoon to lift them onto a plate lined with kitchen paper. Repeat with the remaining spears. After each batch comes out of the butter, grate over the Parmesan. Serve with the mayo.