1. Preheat the oven to 350°F and line a cupcake pan with 12 paper liners.
2. Prepare the flax eggs and set aside to gel for 10 minutes.
3. Finely grate the carrots (I like to do this briefly in the food processor bowl, not with the grating attachment). Press between two layers of paper towel to remove excess moisture.
4. Whisk together the flours, baking powder, baking soda, and spices. Set aside.
5. In a separate bowl, beat together the sugar, oil, almond milk, flax eggs, applesauce/yogurt, and vanilla until well combined.
6. Add the dry ingredients and mix until just combined.
7. Fold in the pressed carrots and chopped walnuts.
8. Spoon into the prepared cupcake pan, filing each 3/4 of the way.
9. Bake for 25-30 minutes at 350°F (they should spring back when lightly pressed in the middle).
10. Cool in the pan for 15 minutes, then transfer to a cooling rack to cool completely.

For the frosting:

1. Beat the cream cheese, butter, vanilla, and salt until creamy and fluffy.
2. Slowly add the powdered sugar, beating on low until incorporated.
3. Turn the mixer up to high and whip for another 10 seconds or so.
4. Frost the cooled cupcakes with a piping bag or offset spatula, and garnish with sugared walnuts of desired.