To make the caramel, place 2/3 cup of sugar in a small saucepan and wet with just enough water to dissolve it (about 1 tablespoon). Heat on medium heat, shaking the pan occasionally, but don't stir. The syrup will boil for a while until it starts to turn brown. Once it has turned brown, remove it from the heat and immediately pour the caramel into 6 ramekins.
In another saucepan, heat the milk with the other 2/3 cup sugar until it reaches a boil. Stir in vanilla. In a sturdy bowl, whisk the eggs for a minute or two. Whisk in the hot milk, slowly at first to avoid coagulating the eggs. Whisk to completely mix.
Pour (or ladle if that works better for you) the milk mixture into the ramekins on top of the caramel. Place all 6 ramekins in a large oven proof dish. Fill the dish with water to about 2/3 the height of the ramekins.
Carefully place in 300° F oven. Bake for 40 minutes to an hour (baking time can depend on the dishes used). The water should not boil during baking. The custard is done when it has set, which you can verify by inserting a knife.
Allow the custard to cool completely, then cover with plastic wrap and refrigerate in their ramekins until serving time.
To serve, run a knife along the outside of each creme, than turn over onto a dessert plate.