In a pan, simmer the anchovies with the garlic and milk over a low heat until nearly dry and the garlic is soft. Allow to cool, then transfer to a blender and blitz to a purée.
Add the egg yolks and mustard. Blitz again, until smooth. Add the lemon juice and zest, and season with a little salt and pepper.
Slowly add the olive and vegetable oils. It will emulsify like a mayonnaise, but will have a creamy consistency from the milk.
Blanch the broccoli in boiling salted water for 1 minute, then refresh in iced water.
Sauté the broccoli briefly in a pan with a little olive oil. Meanwhile, cook the hazelnuts in the remaining olive oil and a little salt until they turn a dark golden colour. Remove with a slotted spoon.
Transfer the broccoli to serving plates, sprinkle over the hazelnuts, spoon over the bagna cauda and sprinkle over the parmesan. Finish each dish with an anchovy.