Place 1 cup cider in a blender.
Place 3 tablespoons grated ginger on several layers of damp cheesecloth. Gather edges of cheesecloth together; squeeze cheesecloth bag over a small bowl. Set 2 tablespoons ginger juice aside, and discard remaining ginger juice and pulp.
Add ginger juice and remaining 1 tablespoon grated ginger to blender, and process until smooth. Add remaining cider and sorbet. Blend until smooth and frothy. Garnish with lemon slices, if desired.