Preheat the oven to 375°F (190°C). Line a 15×10-inch jelly roll pan with parchment paper and lightly grease it.
In a large bowl, beat the eggs and granulated sugar together using an electric mixer on high speed until thick and pale, about 5 minutes.
Add the vegetable oil and vanilla extract to the egg mixture and mix until combined.
In a separate bowl, whisk together the all-purpose flour, baking powder, baking soda, ground cinnamon, ground nutmeg, and salt.
Gradually fold the dry ingredients into the wet ingredients until just combined. Do not overmix.
Fold in the grated carrots and chopped walnuts gently.
Pour the batter evenly into the prepared jelly roll pan and spread it out smoothly.
Bake for 12 to 15 minutes, or until the cake springs back when lightly touched and a toothpick inserted in the center comes out clean.
While the cake is baking, lay a clean kitchen towel on a flat surface and dust it generously with powdered sugar.
Once the cake is done, immediately loosen the edges with a knife and invert the cake onto the prepared towel.
Carefully peel off the parchment paper. Starting from the short side, roll the cake and towel together into a tight spiral. Let it cool completely on a wire rack.
To prepare the filling, beat the softened cream cheese and unsalted butter together until smooth and creamy.
Add the powdered sugar and vanilla extract to the cream cheese mixture and beat until well combined and fluffy.
Once the cake has cooled, gently unroll it and spread the cream cheese filling evenly over the surface.
Roll the cake back up without the towel, making sure the seam is on the bottom.
Wrap the jelly roll in plastic wrap and refrigerate for at least 1 hour before serving.
Before serving, dust the top with additional powdered sugar if desired. Slice and enjoy!