Preheat the oven to 350°F (175°C) and line an 8x8-inch square baking pan with parchment paper.

In a large mixing bowl, whisk together the oat flour, 2 scoops of protein powder, cinnamon, ginger, nutmeg, baking powder, baking soda, and salt.

In a separate medium bowl, whisk the eggs, applesauce, maple syrup, and 1 tsp vanilla extract until smooth.

Fold the finely grated carrots into the wet mixture until evenly distributed.

Combine the wet and dry ingredients, stirring with a spatula until just incorporated; do not overmix.

Pour the batter into the prepared pan, smoothing the top with a spatula.

Bake for 30 to 35 minutes, or until a toothpick inserted into the center of the cake comes out clean.

Remove from the oven and allow the cake to cool completely in the pan, then transfer to a wire rack.

Prepare the frosting by whisking together the Greek yogurt, 1 scoop of protein powder, and 1/2 tsp vanilla extract until creamy and smooth.

Spread the protein frosting evenly over the cooled cake and garnish with chopped walnuts if desired.

Refrigerate the cake for at least 30 minutes before slicing to allow the frosting to set properly.