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Assemble up to 4 hours ahead; cover and chill.
Divide half the sponge fingers among 4 individual serving dishes or glasses, breaking them up to fit snugly over the base of the dishes. Drizzle over half of the cooled espresso.
In a bowl, beat together the crème fraîche, quark, Kahlua and sugar until smooth; spoon half over the sponge fingers. Repeat with another layer of sponge fingers, espresso and crème fraîche mixture. Grate over the chocolate and chill for at least 1 hour.