Preheat oven to 300˚F (150˚C). Prepare 5 (4-ouces) glass jars or ramekins.
For the caramel, in a small saucepan, combine sugar and 1 teaspoon water. Cook over medium heat, stirring often with a high heat resistant rubber spatula or wooden spoon. Sugar will form clumps, but melt eventually into a dark amber color sauce. Add the 1 teaspoon water into the sauce. Be very careful, the caramel will bubble rapidly when water is added. Quickly divide the sauce into the glass jars as the caramel will harden. Don't worry if not every jar have the same amount of sauce and the sauce is not spread evenly at the bottom.
Bring a kettle of water to a boil.
For the pudding, in a medium bowl, whisk together eggs and egg yolk until light foam begins to form. Set aside.
In a small saucepan over medium heat, combine milk, heavy cream, and sugar. Cook until sugar just melted, stirring often. Stir in vanilla extract (or vanilla bean paste). Gradually add all the milk mixture into the yolk mixture, whisking continuously.
Pour the mixture through a sieve into a large measuring cup. Remove bubbles with a piece of wax paper or parchment paper, by pushing it directly on top of the mixture and pulling it away. If needed, repeat one more time.
Divide the mixture evenly into the glass jars. Prepare a water bath. Put the glass jars into a large baking pan or roasting pan. Pour enough boiling water into the pan to come halfway up the sides of the jars. Bake just until the purin is slightly puffed on top, but still jiggling in the center, about 30 to 35 minutes.
Carefully transfer the jars on to a cooling rack and cool until slightly warm. Cool in the fridge for at least 3 hours or until cold. Cover with plastic wrap when cool completely. Serve within 3 days.