Preheat the oven to 375°F (190°C). Line a 10×15 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 until thick and pale, about 5 minutes.
Add the vegetable oil and 1 teaspoon vanilla extract to the egg mixture and mix until combined.
In a separate bowl, whisk together the all-purpose flour, baking powder, ground cinnamon, ground nutmeg, and salt.
Gently fold the dry ingredients into the wet ingredients until just combined.
Fold in the finely grated carrots and chopped walnuts evenly into the batter.
Pour the batter into the prepared jelly roll pan and spread it evenly with a spatula.
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 while rolled.
In a medium bowl, beat the softened cream cheese, unsalted butter, powdered sugar, and 1 teaspoon vanilla extract together until smooth and creamy.
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, wrapping it tightly.
Wrap the rolled cake in plastic wrap and refrigerate for at least 1 hour before slicing and serving.