Heat up the barbecue. To make the garlic mayo, wrap the garlic cloves in a foil parcel, position on a hot spot of the barbecue and cook for 20 mins until really soft. Alternatively, bake in a hot oven for 20-30 mins. Leave to cool, then squeeze the cloves out of their skins and mash with a fork. Mix the garlic purée with the mayonnaise, then chill until ready to serve.
Remove the stalks from the middle of the mushrooms and finely chop them. Heat a drizzle of oil in a pan (on the barbecue or hob), add the stalks and fry for a few mins until golden and soft. Add the peppers, thyme, breadcrumbs, tomato paste, paprika and some seasoning. Cook for 5 mins more, then set aside to cool a little. Rub the mushroom caps with a little oil, season, then top each one with 1⁄4 of the mixture. Can be chilled for up to 1 day.
Meanwhile, heat a little oil in another frying pan (on the barbecue or hob), and add the onions. Cook for 15 mins until soft and golden, then add the sugar, vinegar and some seasoning. Cook for
5 mins more until caramelised and sticky. Can be chilled for up to 2 days.
Put the mushrooms on the barbecue (stuffed side up), close the lid or cover with foil, and cook for 20 mins until soft and cooked through. Be sure to keep an eye on the heat and move to the upper shelf if the bottoms of the mushrooms start to burn. Split the rolls and heat these on the barbecue, too. Spread each roll with some garlic mayo, top with salad leaves, a filled mushroom, some sticky onions and a grating of cheese.