Pulse walnuts in a food processor or blender until finely crushed; transfer to a large bowl. Whisk gluten free flour, bicarbonate of soda, cinnamon, cream of tartar and salt with the walnuts.
Puree carrots in a food processor; transfer to a separate bowl. Whisk orange juice, water, egg and walnut oil with the pureed carrots until smooth; stir into the dry ingredients to make a rich pancake batter. Set pancake batter aside to rest for 5 minutes.
Heat a small amount of oil in a non-stick pan over medium heat. Tilt pan to coat cooking surface with oil.
Stir the pancake batter again with a whisk. Use a ladle to pour out enough batter for one pancake onto the pan.
Cook until the edges start to look dry, 3 to 5 minutes; turn and continue cooking until the other side is browned, 3 to 5 minutes more. Repeat with remaining pancake batter.