Place the raspberries and 2 tablespoons of the honey in a medium bowl and stir gently. Add the yogurt and carefully fold in the berries until thoroughly combined; some berries should remain whole.
Place a large strainer or colander in a medium bowl and line it with 4 layers of cheesecloth or plain white paper towels, allowing the cloth to extend beyond the edge of the strainer. Spoon the yogurt mixture onto the cloth, cover loosely with plastic wrap, and refrigerate for at least 8 hours.
Turn the yogurt cheese onto a serving plate and remove the cloth. Discard the drained liquid. Drizzle the remaining honey over the yogurt cheese. Use the gingersnaps as crackers, topping them with the cheese and extra berries.
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