Preheat the oven to 180C/355F/Gas 4 and preheat the grill to its highest setting.
For the caramelised onions, heat the oil in a frying pan over a very low heat. Add the onions and gently fry for 45 minutes, stirring regularly, until the onions are caramelised, golden-brown and sweet.
Meanwhile, for the mushrooms, place the mushrooms in a baking dish and season with salt and freshly ground black pepper. Sprinkle with the butter and garlic, then pour over the olive oil. Transfer to the oven to bake for 10-12 minutes, until softened slightly.
For the rarebit, place the cheese, mustard, garlic and egg into a bowl and mix together well.
Remove the mushrooms from the oven. Spoon over the caramelised onions and some of the rarebit mixture, pressing this topping into each mushroom.
Transfer the mushrooms to the hot grill and cook until the cheese bubbles, melts and becomes golden-brown.
To serve, place a large cheese-coated mushroom onto each of six plates.
