Preheat the oven to 325 F.
Place 4 ramekins into a small roasting or baking pan.
Add the eggs, zest, 1/4 cup sugar, and salt to a large bowl, whisk until well mixed and creamy. Stopping to scrape the sides and bottom of the bowl.
Add chopped white chocolate to a medium bowl.
Add the heavy cream to a medium saucepan over medium heat, stir and heat until hot and slightly simmering, do not boil.
Remove from the heat and pour over the chopped chocolate, let sit for 1 minute, then stir until fully combined and smooth. Stir in the vanilla extract or vanilla beans.
With the mixer on low speed, gradually add the chocolate mixture to the egg mixture, slowly mixing until fully combined.
Use a spoon to scoop and discard any foam that has accumulated on top.
Place 1 tablespoon of the prepared fig compote in the bottom of each ramekin. Evenly divide the crème brûlée mixture between the 4 ramekins on top of the compote.
Pour hot water into the roasting pan to come halfway up the sides of the ramekins.
Bake in the preheated oven for 45-50 minutes. When the crème brûlée is done, it will look set but will still have a slight jiggle in the middle.
Remove the roasting pan from the oven and allow the crème brûlée to cool in the pan. Once cooled, remove ramekins from the roasting pan and cover with plastic wrap. Transfer to the fridge for at least 2 hours and up to 3 days before serving.
When ready to serve, remove from the refrigerator and allow to come to temp for 30 minutes. Sprinkle half a tablespoon of sugar over the top of each ramekin and use a torch to melt the sugar to form a crispy sugar layer. If you do not have a torch, broil the crème brûlée in the oven to melt the sugar, but keep a close eye on it to ensure it doesn't burn. Allow the crème brûlée to sit for 10 minutes before serving.