Heat the oven to 300°F. In a large mixing bowl, lightly whisk the egg yolks until smooth. Set aside.
In a medium, heavy-based saucepan, heat the brown sugar, milk, and cream over medium-high heat, stirring to dissolve the brown sugar. Scald the mixture (don’t boil it) and remove it from the heat.
While the cream mixture is heating, dissolve the sugar and water over low heat in a large, heavybased saucepan. Increase the heat to high and boil the sugar. If the sugar starts to sputter, use a pastry brush dipped in water to wash down the sides of the pan. As soon as the sugar turns dark amber (like an old penny), carefully and slowly pour the hot cream into it, stirring the mixture. (The caramel will bubble as you add the cream, so use a long wooden spoon or a whisk to combine them.)
Gently and slowly whisk this mixture into the egg yolks. Stir in the salt and vanilla. Strain and cool in an ice bath or refrigerate until cool.
Skim off any air bubbles from the pudding mixture and divide the cool mixture among six 6-oz. ramekins. Set the ramekins in a shallow roasting pan and fill the pan with hot water until it reaches halfway up the sides of the ramekins. Cover the pan with foil.
Bake the puddings until just set. Begin checking after 1 hour. When gently shaken, they should no longer look liquid; instead, the custard will move as one mass. Allow the puddings to cool to room temperature in the water. Refrigerate until cool, cover, and then chill for several hours or overnight before serving.