Preheat your oven to 350°F (175°C). Prepare an 8x8-inch tray by greasing or lining with parchment paper.

In a medium bowl, whisk together flour, baking powder, baking soda, cinnamon, nutmeg, and salt until well combined.

In a large bowl, beat together eggs, brown sugar, granulated sugar, applesauce, olive oil, and vanilla extract. Fold in grated carrots.

Gently fold the dry ingredient mixture into the wet mixture until just combined.

Pour the batter into the prepared tray and bake for 25–30 minutes, checking for doneness with a toothpick.

Let the cake cool completely in the pan on a wire rack.

For the frosting, beat together cream cheese, butter, vanilla extract, and powdered sugar until creamy.

Spread the frosting evenly over the cooled cake and garnish as desired.

Slice into squares for serving.