Heat the cream, milk and sugar in a heavy saucepan to almost boiling. In a separate bowl, beat the egg yolks together well.
Gradually whisk the heated mixture into the eggs, then return the mixture to the saucepan. Cook over medium heat, stirring constantly with a wooden spoon until the custard coats the back of the spoon, 3 to 4 minutes; remove from heat. Stir in the vanilla.
Pour the custard into 6 individual custard dishes or into one shallow baking dish, about 9 inches across. Set the dishes or dish in a large pan and place on the center rack of the oven. Carefully pour hot water into the outer pan to come level with the custard.
Bake until the center of the custard is set, 35 to 45 minutes. When done, remove the custard from the water bath and cool. Cover and chill.
A few hours before serving, preheat the broiler.
Sift the brown sugar evenly over the top of the individual custards, spreading it to the edges. Set the custards on a baking sheet and place under the broiler as close to the heat as possible. Broil until browned but not burned, about 1 1⁄2 minutes. Watch closely. Remove and chill.