Place the eggs in a small saucepan, cover with cold water, and bring to a simmer. Cook for 10 minutes to ensure the yolks are completely firm. Drain immediately and submerge them in ice-cold water to cool.
Peel the cooled eggs and separate the whites from the yolks. (You can save the whites to slice up into your final salad). Place the hard-boiled yolks into a medium mixing bowl.
Add the mustard powder, caster sugar, salt, and cayenne pepper to the yolks. Using the back of a sturdy spoon or a small fork, mash and cream everything together vigorously until it forms a perfectly smooth, lump-free paste.
Pour in the double cream a little bit at a time, whisking constantly with a balloon whisk. Ensure each splash is fully smooth before adding the next. The mixture will begin to look rich and velvety.
Slowly drizzle in the vinegar while continuing to whisk. The acid in the vinegar will react with the cream and the egg yolks, naturally thickening the dressing into a gorgeous, spoonable sauce.