Mix eggs, milk, sugar, and vanilla extract until smooth. Strain the mixture twice to create an extra silky texture.

Pour into heatproof coffee cups and cover tightly with aluminum foil. Place the cups in a pot and add water around them until it reaches about the same height as the custard inside.

Cover with a lid and cook over medium heat until the water gently boils. Turn off the heat and let sit, covered, for 20 minutes. The center should jiggle slightly but not appear liquid.

Chill in the refrigerator for 2 hours (or 1 hour in the freezer) until fully set.

To make the caramel sauce, heat sugar and 1 tablespoon of water in a nonstick pan over medium heat. Do not stir. Gently swirl the pan if needed, but avoid using a spatula to prevent crystallization.

Cook until the sugar turns a deep amber brown. Carefully pour in the hot water all at once (it will splatter), then stir until smooth and fully liquid.

After the custard has chilled, pour the caramel sauce over the top and serve.