Heat the oven to 350 F (175 C). Grease two 8-inch round cake pans and line the bottoms with rounds of parchment paper.

Whisk the flour, granulated sugar, brown sugar, baking powder, baking soda, salt, cinnamon, ginger, and nutmeg together in a large bowl until no streaks of spice or flour remain.

In a second bowl, whisk the oil, oat milk, apple cider vinegar, and vanilla until the mixture looks smooth and slightly thickened. The vinegar will make the milk look a little curdled, and that's exactly what you want.

Pour the wet ingredients into the dry and stir with a spatula just until the last dry patches disappear, about 15 to 20 strokes. Fold in the grated carrots and the nuts. Stop as soon as everything is evenly combined, because a batter that's stirred too long bakes up tough.

Divide the batter evenly between the two pans, about 2 1/2 cups per pan, and smooth the tops with the spatula.

Bake for 28 to 32 minutes, until the centers spring back when you press them gently and a toothpick inserted in the middle comes out clean or with a few moist crumbs.

Cool the cakes in the pans for 15 minutes, then run a knife around the edges, turn them out onto a wire rack, and let them cool completely before frosting.

For the frosting, beat the softened vegan cream cheese and vegan butter with an electric mixer until smooth and fluffy, about 2 minutes. Add the powdered sugar one cup at a time on low speed, then beat in the vanilla, salt, and cornstarch until the frosting is thick and holds a soft peak. If it feels loose, chill it for 20 minutes before using.

Place one cooled layer on a serving plate and spread about a cup of frosting over the top. Add the second layer, then frost the top and sides with the rest. Refrigerate for at least 1 hour so the frosting sets before you slice.