Preheat the oven to 350°F (175°C). Lightly coat an 8-inch square baking pan with cooking spray and line the bottom with parchment paper.
Blend the rolled oats until they form a fine, flour-like powder.
In a large bowl, whisk together the blended oats, baking powder, baking soda, cinnamon, and salt.
In a separate bowl, whisk the eggs, Greek yogurt, applesauce, granulated sweetener, and vanilla until smooth.
Add the wet ingredients to the dry ingredients and fold until combined. Fold in the grated carrots and chopped walnuts.
Let the batter rest for 5 minutes, then spread it evenly in the prepared pan.
Bake for 28–35 minutes, checking the center with a toothpick. It should come out without wet batter, although a few moist crumbs are fine.
Cool the cake in the pan for 15–20 minutes, then transfer it to a cooling rack and let it cool completely.
Beat the softened cream cheese until smooth. Add the Greek yogurt, powdered sweetener, and vanilla, then mix until creamy.
Spread the frosting over the cooled cake. Refrigerate for 20–30 minutes to help the topping firm up, then cut into 9 equal squares.