Preheat your oven to 325°F. Boil 5–6 cups of water for a water bath.
In a medium saucepan, combine the heavy cream, vanilla bean paste, and brown sugar. Heat over medium, stirring often, until the sugar is fully dissolved—about 5 minutes. Do not let it boil.
In a separate bowl, lightly whisk the egg yolks and salt to combine.
Slowly pour the warm cream mixture into the yolks, whisking constantly to avoid curdling.
Place a 3.5-quart braiser in a large roasting pan. Pour the custard into the braiser. If bubbles form, skim with a spoon or strain through a fine-mesh sieve.
Pour the boiling water into the roasting pan to create a water bath, filling it halfway up the sides of the braiser. Carefully transfer to the oven.
Bake for 40–45 minutes, until the edges are set but the center still wobbles slightly.
Remove the braiser from the water bath and cool to room temperature. Then refrigerate for at least 4 hours or overnight.
Right before serving, sprinkle an even layer of white sugar over the top. Use a kitchen torch to caramelize the sugar, moving in circles until the top is golden and crackly.
Chill again for at least 30 minutes to re-set the top. Serve cold, straight from the braiser with spoons for family style.