Butter a 9 by 13-inch glass baking dish. Arrange half of the brioche along the bottom of the dish, and cut or tear the slices to fit. Spread 3/4 cup of the jam on top and cover with the remaining bread.
Whisk the eggs, sugar, 2 1/2 cups of the milk, and the vanilla together; pour over the bread. Press to soak and brush the top with 4 tablespoons of the melted butter. Cover with foil and bake for 15 minutes. Remove the foil and continue baking for 15 minutes longer or until the pudding is set. Remove from the oven.
Preheat the broiler. In a bowl, whisk the remaining 1 tablespoon of milk with the confectioners' sugar. Stir in the remaining melted butter and jam until the mixture is smooth. Spread the mixture over the top of the pudding, place under the broiler, and broil until the glaze is golden. Serve hot or warm, garnished with fresh strawberries.