For the brine, put all the ingredients in a medium non-reactive saucepan with 1 litre/1¾ pint water. Slowly bring to the boil, stirring as you go to dissolve the salt and sugar. Remove from the heat and leave to cool.
Place the beef bavette in the brine and leave, covered, for 1 hour. Drain and transfer to a clean bowl. Add the extra virgin olive oil, garlic, chilli, lemon zest and thyme, then leave to marinate for at least 1 hour.
Light the barbecue so that it is ready when the steaks have finished marinating.
Meanwhile for the salsa, mix all the ingredients together, seasoning to taste and adding the lemon juice at the last minute.
Remove the beef from the marinade, season with salt and pepper and place on the grill in the direct heat zone. Keep a close eye on the steaks - bavette cooks quickly as it is quite thin, and it shouldn’t be cooked past medium-rare otherwise it’ll be tough. Grill for 2 minutes on each side to char, then move to the cooler edge of the barbeque to rest for a couple of minutes.
Toss the onions with a little olive oil, season with salt and pepper and place directly on the grill in the direct heat zone and cook for 3-4 minutes until charred and tender.
Thickly slice the steaks and serve with the grilled onions and salsa cruda.