Cut the mushrooms into thin slices. Chop the green pepper.
Heat the oil in a frying pan and add the mushroom and pepper, then the garlic and oregano. Season with salt and pepper.
Turn down the heat, put a lid on and gently sauté for 5 minutes, then add the vinegar and tomatoes and sauté for another 5 minutes. Add the chopped parsley to the mushrooms and keep warm under a lid.
Meanwhile, in another pan fry the eggs ( or poach them in boiling water).
Cut the bagels in half and lightly toast them. You can spread some red pesto or other paste on them, if you like.
Spoon the mushroom mixture on one of the bagel halves, then add the egg and top with the other half of the bagel.