Preheat the oven to 300 degrees F (150 degrees C).

Whisk egg yolks, 4 tablespoons white sugar, and vanilla extract in a mixing bowl until thick and creamy; set aside.

Pour cream into a saucepan and stir over low heat until it almost comes to a boil.

Whisk cream into egg yolk mixture; beat until combined.

Pour 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.

Pour into a shallow heat-proof dish or individual ramekins.

Bake in the preheated oven until custard is set but slightly jiggly, about 30 minutes. Let cool to room temperature, then refrigerate until chilled, at least 1 hour or overnight.

Turn the oven to broil.

Combine remaining 2 tablespoons white sugar and brown sugar in a small bowl. Sprinkle mixture evenly over custard.

Place dish under the broiler until sugar melts and caramelizes, about 2 minutes. Watch carefully to avoid burning.