Combine 1/2 cup cream and white chocolate in a heavy saucepan; cook over low heat, stirring constantly, 2 to 3 minutes or until chocolate is melted. Remove from heat, and stir in vanilla and remaining 1 1/2 cups cream.
Whisk together egg yolks and 1/4 cup sugar until sugar is dissolved and mixture is thick and pale yellow. Add cream mixture, whisking until well blended. Pour mixture through a fine wire-mesh strainer into a large bowl.
Toss banana slices with 1/3 cup sugar. Melt butter in a large nonstick skillet over medium-high heat. Add bananas, and cook 1 to 2 minutes on each side or until lightly browned. Line bottoms of 6 (4-oz.) ramekins evenly with banana slices. Pour cream mixture evenly into ramekins; place ramekins in a large roasting pan. Add water to pan to depth of 1/2 inch.
Bake at 300° for 45 to 55 minutes or until edges are set. Cool custards in pan on a wire rack 25 minutes. Remove ramekins from water bath; cover and chill 8 hours.
Sprinkle 1 1/2 to 2 tsp. remaining sugar evenly over each ramekin. Fill a large roasting pan or 15- x 10- x 1-inch jelly-roll pan with ice; arrange ramekins in pan.
Broil 5 inches from heat 3 to 5 minutes or until sugar is melted and caramelized. Let stand 5 minutes. Garnish, if desired.