Add the yogurt and buttermilk in a blender. Rinse the cucumber, slice a small piece, and set aside for garnish. Coarsely chop the remaining cucumber, then add the blender. Peel the garlic and rinse the dill. Reserve some dill for garnish. Add the remaining dill and the garlic to the blender. Puree the buttermilk mixture in the blender on a high setting.
Season the mixture to taste with salt and pepper. Pour into glasses and garnish with the sliced cucumber and the reserved dill.