In a small bowl, combine the cheese, mayonnaise and garlic; spread over each muffin half. Place on a baking sheet. Broil 4-6 in. from the heat for 2-3 minutes or until golden brown. Set aside.
Place 2-3 in. of water in a large skillet with high sides; add vinegar. Bring to a boil; reduce heat and simmer gently. Break cold eggs, one at a time, into a custard cup or saucer; holding the cup close to the surface of the water, slip each egg into water. Cook, uncovered, until whites are completely set and yolks begin to thicken (but are not hard), about 4 minutes.
Meanwhile, in a large skillet, saute mushrooms and shallot in oil until tender. Stir in the cream, horseradish, mustard, white pepper and cayenne. Bring to a boil. Reduce heat; simmer, uncovered, for 3-5 minutes or until slightly thickened, stirring constantly.
With a slotted spoon, lift each egg out of the water. Top each muffin half with an egg and 2 tablespoons sauce. Sprinkle with chives. Serve immediately.
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