Whisk milk and 5 tablespoons sugar together in a saucepan over low heat; heat just until warmed and sugar has dissolved. Remove from heat and whisk eggs into milk mixture. Pour caramel syrup into an 8x8-inch baking dish. Pour egg-milk mixture over syrup. Place pan into a larger baking dish and pour enough water into outer pan to reach about 1/2 inch up the sides of the inner pan.