Beat egg yolks, 4 tablespoons white sugar and vanilla extract in a mixing bowl until thick and creamy.
Pour cream into a saucepan and stir over low heat until it almost comes to boil. Remove the cream from heat immediately. Stir cream into the egg yolk mixture; beat until combined.
Pour cream mixture into the top pan of a double boiler. Stir over simmering water until mixture lightly coats the back of a spoon, about 3 minutes. Remove mixture from heat immediately and pour into a shallow heat-proof dish.
Bake in preheated oven for 30 minutes. Remove from oven and cool to room temperature. Refrigerate for at least 1 hour or overnight.
Preheat oven to broil.
In a small bowl combine remaining 2 tablespoons white sugar and brown sugar. Sift this mixture evenly over custard. Place dish under broiler until sugar melts, about 2 minutes. Watch carefully so as not to burn.
Remove from heat and allow to cool. Refrigerate until custard is set again.