Place apple in a plastic microwave-safe bowl; pour in enough water to cover 1/4-inch of the bottom of bowl. Partially cover bowl with a lid or paper towel. Microwave in 1 minute intervals until apple is softened, 2 to 3 minutes. Freeze apple in the same container with water until solid, 2 hours to overnight.
Blend frozen apple, almond milk, and pumpkin puree in a blender until smooth; add brown sugar and pumpkin pie spice. Blend until smooth. Add ice and blend until smooth.