Step 1: Preheat oven to 350°F (175°C) and line an 8×8-inch baking pan with parchment paper.
Step 2: In a medium bowl, whisk together flour, baking powder, baking soda, cinnamon, nutmeg, and salt.
Step 3: In a large bowl, beat eggs and granulated sugar until light and fluffy, about 2 minutes.
Step 4: Gradually mix in vegetable oil and vanilla extract until well combined.
Step 5: Fold in the dry ingredients until just incorporated, then stir in grated carrots and walnuts.
Step 6: Spread batter evenly into the prepared pan and bake for 25-30 minutes, or until a toothpick inserted comes out clean.
Step 7: Let bars cool completely in the pan on a wire rack.
Step 8: For the frosting, beat cream cheese and butter together until smooth, then gradually add powdered sugar and vanilla until creamy.
Step 9: Spread frosting over cooled bars, cut into squares, and serve.