Heat a little of the olive oil and sauté the mushrooms for a minute or two and season.
Set aside. Bring a shallow pan of water to simmer, add the vinegar and salt. Break one egg onto a saucer, stir the water to create a whirlpool, tip in the egg and leave to cook for 3 minutes. Using a slotted spoon, lift on to a plate. Repeat with the remaining eggs. Keep the water hot.
Heat the grill to high, arrange the mushrooms on the untoasted side of the bread, cover with the tomatoes and then the cheese, grill for 5-6 mins until the cheese has just melted. Place the toast on warmed plates. Slide all the eggs back into the hot water and leave for 30 seconds, lift out and drain on kitchen paper, then put one on top of toast. Season and serve.