Mix 1 tablespoon water into 1/4 cup sugar in a heavy saucepan over medium heat until sugar is completely moistened. Bring to a boil, reduce heat to medium-low, and cook until the sugar and water form a golden brown syrup, about 5 minutes. Do not stir. Remove from heat and let boiling subside. Carefully stir 1 tablespoon hot water into caramel syrup.
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.
Bake in the preheated oven until custard has set, about 45 minutes. Let cool for about 10 minutes; chill in refrigerator before serving.
