Preheat the oven to 350°F (175°C) and grease the bottom of an 8-inch cake pan.
Using a grater, finely shred the carrots and measure out 2 cups.
In a large mixing bowl, whisk together eggs, sugar, ground cinnamon, ground nutmeg, and vegetable oil until the mixture is smooth and slightly thickened, about 2-3 minutes. Watch for the mixture to become glossy and uniform.
Add the grated carrots to the wet mixture and fold gently until evenly combined.
In a separate bowl, sift together flour, baking powder, and salt. Gradually add the dry ingredients to the wet mixture, folding until just incorporated. The batter should be thick but pourable.
At this stage, the batter should be thick but still pourable. Stop mixing as soon as everything is combined to keep the cake tender.
Pour the batter into the prepared cake pan, smoothing the top with a spatula. Bake in the preheated oven until the surface is golden brown and a toothpick inserted into the center comes out clean, approximately 40 minutes. Watch for the edges to pull slightly away from the pan and for a pleasant aroma to develop.