The first step is not essential, but does give an extra measure of elegance. Spread the brown sugar out, breaking all the lumps, on a big plate. Put it into the microwave oven for 10 minutes at 10 percent power, then let it cool for 30 minutes. This will remove the excess moisture from the brown sugar and keep it from turning to syrup when you blast it later.
Combine 1/4 cup of cream and the egg yolks in a metal bowl, and whisk to blend well. Stir in the sugar until nearly dissolved.
Put the rest of the cream into a small saucepan and heat it over medium heat until wisps of steam start appearing. (Don't boil even a little.)
Add the vanilla to the warm cream. Stir, then pour the warm cream slowly into the metal bowl while whisking.
Strain the custard through a fine sieve into a large measuring cup. Pour the custard into the baking dishes.
Pour hot water into the pan until it's halfway up the sides of the baking dishes. Put the pan into the preheated 325-degree oven and bake for 30 minutes. Depending on what dishes you're using, it may take as long as another 15 minutes. The custard should be set but not solid.
Remove the dishes from the pan and set out to cool for a half hour, then refrigerate for at least three hours, or as long as a day.
When ready to serve, preheat the broiler. (Or the broil feature of your toaster oven, which works better for this than you might imagine.) Sprinkle enough brown sugar on top of each custard to completely cover, and run them under the broiler for about 30 seconds--until the sugar melts. You might want to turn the dishes so that this happens uniformly.