Position rack in center of oven and preheat to 375 degrees. Whisk yolks and sugar. Combine cream and coffee in small saucepan and bring to boil over medium heat, stirring constantly. Slowly whisk hot cream into yolks. Whisk in kahlua and vanilla. Pour mixture into 4 round broiler proof baking dishes about 3 inches wide and 1 inch deep.
Place dishes in shallow pan and add boiling water to come halfway up sides of dishes. Bake until tops are set, 30 minutes. Remove from water bath. Cool to room temperature. Cover and chill overnight. Preheat broiler. Press brown sugar through strainer onto custard. Broil 5 to 6 inches from heat until sugar liquifies and begins to caramelize. Be careful to avoid burning. Refrigerate 1 hour before serving.
