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 with an electric mixer on high speed for 5 minutes, until thick and pale.
Add the vegetable oil and vanilla extract to the egg mixture and beat until combined.
In a separate bowl, whisk together the flour, baking powder, baking soda, cinnamon, nutmeg, and salt.
Gradually fold the dry ingredients into the wet ingredients using a spatula until just combined.
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.
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 a short side, roll the cake and towel together into a tight spiral.
Allow the rolled cake to cool completely on a wire rack.
To make the frosting, beat the softened cream cheese and butter together until smooth and creamy.
Gradually add the powdered sugar and vanilla extract, beating until the frosting is light and fluffy.
Unroll the cooled cake gently and spread the cream cheese frosting evenly over the surface.
Roll the cake back up without the towel and place it seam-side down on a serving plate.
Refrigerate the jelly roll for at least 1 hour before slicing and serving.