Smoky mushroom burgers with roasted garlic mayo

Name: Smoky mushroom burgers with roasted garlic mayo
Ingredients: 4 large flat mushrooms; 1 tbsp olive oil plus extra for frying; 2 roasted red peppers (from a jar), finely chopped; ½ small pack thyme leaves picked and chopped; 50g fresh breadcrumb; 1 tbsp sundried tomato paste; 2 tsp smoked paprika; 3 red onions thinly sliced; 1 tbsp golden caster sugar; 1 tbsp sherry vinegar; 3 garlic cloves unpeeled; 50g good-quality mayonnaise; 4 crusty bread rolls (we used ciabatta); salad leaves; 25g cheddar or manchego grated