Preheat oven to 350º and prepare small loaf pans by spraying with olive oil. We used two small pans, 5" by 2 1/2".
In a small bowl, combine ground flax seed and water and mix together using a fork. Set aside while it sets up. It will become gelatinous and egg-like.
In a large bowl, whisk together flour, baking soda, baking powder, sugar, salt, and pumpkin pie spice.
Combine flax eggs, mayonnaise, and pumpkin in another bowl and mix well.
Add wet mixture to dry mixture and stir together until well combined.
Fill the loaf pans with the mixture and pop them into the oven. Bake 30 minutes or until you can insert a toothpick near the center and remove it cleanly.