Saute shallots and garlic in butter in a skillet over medium-high heat until tender. Add mushrooms; saute until liquid evaporates. Stir in 1/4 teaspoon salt and black pepper. Arrange fillets in a greased 11- x 7- x 1 1/2-inch baking dish. Sprinkle with 1/2 teaspoon salt and 1/4 teaspoon white pepper. Pour champagne and lemon juice over fillets. Cover and bake at 400° for 10 minutes. Drain, reserving liquid. Strain liquid through cheesecloth. Cook strained liquid in a saucepan over high heat until liquid is reduced to 1 cup.
Melt 1/4 cup butter in a heavy saucepan over low heat; add flour, stirring until smooth. Cook 1 minute, stirring constantly. Add whipping cream; cook over medium heat, stirring constantly, until mixture is thickened and bubbly. Stir in remaining 1/4 teaspoon salt and 1/4 teaspoon white pepper. Stir about one-fourth hot mixture into egg yolks; add to remaining hot mixture, stirring constantly.
Spoon mushroom mixture over fillets; top with sauce, and sprinkle with cheese. Bake, uncovered, at 375° for 15 minutes or until thoroughly heated. Broil 5 1/2 inches from heat (with electric oven door partially opened) 2 minutes or until lightly browned. Serve immediately.