Preheat the oven to 425°F (220°C). Grease four ramekins thoroughly using butter, then lightly dust the interiors using flour. Arrange them on a baking tray so transferring them becomes easier.

Combine the chopped white chocolate and butter inside a heatproof bowl. Warm gently over barely simmering water or microwave using short intervals until completely smooth. Stir gently until glossy, then allow the mixture to cool slightly.

Whisk the whole egg, yolks, powdered sugar, and vanilla until pale, creamy, and slightly thickened. This creates a light batter that helps support the delicate exterior.

Fold the melted chocolate mixture into the egg mixture using a spatula until evenly blended. Sprinkle in the flour gradually, mixing only until no dry streaks remain.

Divide the batter evenly among the prepared ramekins, filling each about three-quarters full. Place the tray onto the center oven rack and bake until the edges appear set while the middle still looks slightly soft.

Remove the cakes and let them rest for one minute. Carefully loosen the edges using a thin knife, invert each ramekin onto a dessert plate, lift away the mold, and serve immediately while the creamy vanilla center remains warm and flowing.