Break up the sponge fingers and dip them in the cold coffee. Transfer to a bowl.
Add the custard and halved raspberries to the soaked sponge fingers and gently fold together. Spoon the mixture into 4 serving glasses and top with the whole raspberries.
Melt the white chocolate in a heatproof bowl over a pan of simmering water (or melt in the microwave). Drizzle over the top of each tiramisu then place in the fridge for 1 hour to chill.