Bring a large pot of water to a boil. Cook carrots in the boiling water until very tender, about 15 minutes. Drain and transfer to a blender. Cover and hold lid down; pulse a few times before leaving on to blend. Puree in batches until very smooth. Set aside to cool.
Preheat oven to 400 degrees F (200 degrees C).
Beat eggs in a large bowl. Add carrots, evaporated milk, brown sugar, honey, salt, cinnamon, nutmeg, cloves, and allspice; beat until smooth. Fit pie crust into a 9-inch pie dish and pour carrot mixture into pie crust.
Bake in the preheated oven until a knife inserted into the center of the pie comes out clean, 40 to 45 minutes.