Fry the mushrooms gently in melted butter until tender and golden brown. Season with salt and pepper. Finish with shallots and chives. Fry the bread crispy in oil and drain on paper.
Break the eggs carefully into 6 individual cups. Add 2 teaspoons of vinegar to each cup. Let settle for 10 minutes. Heat water and 1 tablespoon vinegar in a saucepan. When it starts boiling move the part off the fire, whisking to create a vortex. Gently add the eggs one by one inwards from the edge. Return the pan to the heat and let it bubble gently. Simmer the eggs for 3 minutes. Lift out of the hot liquid.
Top croutons with sautéed mushrooms, poached eggs, dollop of caviar and chopped dill. Serve in a wide soup bowl.