Preheat oven to 350°F (175°C) and line an 8-inch square baking pan with parchment paper.
In a large mixing bowl, whisk together the eggs, 0.5 cup Greek yogurt, applesauce, 0.33 cup maple syrup, and vanilla extract until homogenous.
In a separate bowl, sift together the oat flour, 60g protein powder, cinnamon, ginger, nutmeg, baking powder, baking soda, and salt.
Gradually incorporate the dry ingredients into the wet mixture using a spatula, stirring until just combined.
Gently fold in the finely grated carrots until evenly distributed throughout the batter.
Transfer the batter to the prepared baking pan and smooth the top with a spatula.
Bake for 30 to 35 minutes or until a skewer inserted into the center comes out clean.
Remove the cake from the oven and allow it to cool completely on a wire rack before frosting.
Prepare the protein frosting by whisking together 1 cup Greek yogurt, 30g protein powder, and 1 tablespoon maple syrup until smooth and thick.
Spread the frosting evenly over the cooled cake and refrigerate for 30 minutes to set before slicing into 8 portions.