Combine pumpkin, ginger, cinnamon, and cloves in a saucepan; cook and stir occasionally, about 5 minutes. Transfer to a bowl; place in freezer to cool quickly.
Soften ice cream in a microwave oven in two 15-second intervals. Place in a large bowl; add pumpkin mixture. Stir until color is uniform. Fold in gingersnaps until evenly distributed. Freeze until firm, about 30 minutes.