Pulse walnuts in a food processor or blender until finely crushed; transfer to a large bowl. Whisk oat flour, baking 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 and hearty batter. Set batter aside to rest for 5 minutes.
Heat a small amount of canola oil in a nonstick skillet over medium heat. Tilt skillet to coat cooking surface with oil.
Stir the batter again with the whisk. Use a 1/4-cup measuring cup to spoon out the batter, but do not fill the measuring cup. Drop small amounts of the batter into the skillet.
Cook pancakes 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.