Using cloves of confit garlic not only gives a lovely mellow garlic flavour, but also helps create a thicker, creamier mayonnaise. Traditionally, aioli was made without egg, making it more of a sauce than a mayonnaise and lacking the yolky yellow colour.

Peel the garlic cloves and cook very slowly in the sunflower oil for 30 minutes. Make sure it doesn’t brown. Cool to room temperature.
Place garlic, egg yolk and mustard in a small food processor, Blitz and slowly add the oils.
When nearly all the oil has been incorporated, add the lemon/vinegar, sugar, salt and pepper and finish adding the oil.
Scrape out into a bowl or jar.
If, by any chance, it splits, you can start again with another egg yolk and add the curdled mix, as with the oil, but I’d be inclined to cut my losses and start again from scratch.