Preheat the oven to 180°C/350°F/gas 4 or get your barbecue nice and hot!
Wrap the red onion in kitchen foil and cook in the oven, or directly in the coals, for 2 hours until soft.
When your onions have 30 minutes cooking time left, make your mayo: separate and whisk the egg yolk until fluffy, then gradually whisk in both the oils – a tiny bit at a time – until thick and emulsified (loosen with a little squeeze of lemon if it gets too thick).
Season to perfection with a pinch of sea salt and a little more lemon juice. Pick the basil leaves into a pestle and mortar, bash until they make a paste and stir through the mayo.
Season the mince to perfection and mix together well. Divide into 4 even-sized pieces and shape into flat patties. Cut the mozzarella in half and place on top of 2 of the patties. Place the remaining 2 patties on top and seal the edges firmly with your fingertips, so the mozzarella is encased inside, then gently shape into a neat burger.
When the onions are cool enough to handle, peel off the skin and mash with a fork, discarding the root. Drizzle over the balsamic vinegar, season to perfection and mix well.
Get your BBQ or grill nice and hot. Add the burgers and cook for 8 to 10 minutes, turning regularly.
Crisp up the mortadella on the grill bars, and shape into a circle with your tongs. Slice the buns and toast on the BBQ. Thickly slice the tomato.
Spoon 1 tablespoon of the basil mayo over the bottom of the bun, add a slice of tomato, then a slice of mortadella and then some rocket. Top with the burger and a dollop of sticky balsamic onions. Pop the lid on and tuck in!