4 ounces white chocolate, broken into small pieces
½ teaspoon pure vanilla extract
¼ cup coffee-flavored liqueur (such as Kahlua®)
Preheat oven to 300 degrees F (150 degrees C).
Whisk egg yolks and 1/3 cup white sugar together in a bowl until smooth.
Bring 1 3/4 cup plus 2 tablespoons heavy cream to a simmer in a saucepan over medium-high heat, about 5 minutes. Add white chocolate. Remove from heat and whisk until chocolate is melted.
Spoon 2 tablespoons of cream mixture into egg yolks, whisking constantly to prevent eggs from scrambling. Pour remaining cream mixture over egg yolks, whisking until custard is thick and smooth. Stir in vanilla extract and coffee liqueur.
Pour custard into four 6-ounce ramekins. Arrange ramekins in a 9×13-inch baking pan. Pour in enough warm water to come 1 1/2 inches up the sides of the ramekins.
Bake in the preheated oven until custard is set, about 45 minutes. Cool to room temperature, about 30 minutes. Cover with plastic wrap and refrigerate until chilled, about 2 hours.
Set oven rack about 6 inches from the heat source and preheat the oven's broiler.
Arrange ramekins on a baking sheet. Sprinkle 1 teaspoon white sugar over each custard.
Broil in the preheated oven until sugar is melted and crispy, about 2 minutes. Chill before serving, 5 to 10 minutes.