Heat the oil in a frying pan. Add the mushrooms and cook over a high heat for 3–4 minutes until golden and tender. Reduce the heat and stir in the soft cheese and milk to make a creamy sauce, adding a little more milk if necessary.
Meanwhile, halve and toast the bagels under a preheated grill. Spoon the creamy mushrooms on top and sprinkle with the Cheddar cheese to serve.