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.
