Position racks in upper and lower thirds of oven; preheat to 350 degrees F (175 degrees C). Line 2 baking sheets with parchment paper.

Whisk grated carrot, peanut butter, applesauce, pumpkin puree, water, and eggs together in a large bowl until thoroughly combined. Stir in 2 1/2 cups flour and cinnamon with a sturdy spoon until incorporated. Add remaining flour, 2 tablespoons at a time, until dough holds together nicely.

Roll dough on a lightly floured surface to an even 3/8-inch thickness. Cut with 3/4-inch cookie cutters and place on the prepared baking sheets about 1/2-inch to 3/4-inch apart.

Bake 2 sheets in the preheated oven for 6 minutes. Switch racks and continue baking until edges are firm and golden brown (tops may still appear a bit puffy), another 6 to 7 minutes. Remove from the oven and allow to cool on the baking sheets for 3 to 5 minutes before removing to a wire rack to cool completely.