Preheat oven to 300 degrees F (150 degrees C). Move a rack to the bottom of the oven.
Whisk the egg yolks, sugar, and vanilla together in a stainless steel bowl until smooth. Bring the cream just to a simmer in a saucepan over medium-low heat; pour into the egg yolk mixture, whisking continually until combined.
Bring a couple of inches of water to a simmer in an oven-safe saucepan over medium low heat; set the bowl with the cream mixture over the saucepan to form a double boiler.
Place the entire double boiler into the preheated oven; bake until the mixture sets, about 30 minutes.
Chill in refrigerator 2 hours.
Preheat the oven’s broiler and set the oven rack about 6 inches from the heat source.
Combine the brown sugar with 2 tablespoons of white sugar in a bowl; sprinkle evenly over the cold creme.
Heat under the preheated broiler until the sugar melts and browns, 2 to 3 minutes; return immediately to the refrigerator and chill until the sugar is set, about 15 minutes.