In a large bowl, combine the eggs, milk and vanilla. Gently stir in bagels. Transfer to a greased 13-in. x 9-in. baking dish; let stand for 15 minutes or until bagels are softened. Meanwhile, place butter and brown sugar in a small saucepan; cook and stir over medium-low heat until sugar is melted. Pour over baking dish., Bake, uncovered, at 350° for 45-50 minutes or until the top is puffed and center is almost set. Let stand for 15 minutes. Serve warm.