Heat the oven to 175°C (350°F). Grease two 20-cm (8-inch) round cake pans, place parchment paper on the bottoms, and lightly dust the sides with flour. Grate the carrots finely. If they release a lot of moisture, gently squeeze them, leaving enough moisture to keep the cake tender.

Add flour, baking powder, baking soda, cinnamon, ginger, nutmeg, and salt to a large bowl and whisk well. In a separate bowl, whisk the eggs and sugar until blended. Pour in the oil and vanilla, then stir until smooth. Slowly mix the dry ingredients into the wet mixture. Add the grated carrots and fold them through using a spatula. Stop mixing as soon as the batter comes together.

Pour the batter evenly into the two prepared pans and level the surfaces. Bake for 30–35 minutes, checking the centers with a toothpick. The cakes are ready when the toothpick comes out clean or has a few moist crumbs attached. Leave the cakes in their pans for 10–15 minutes, then move them to a wire rack. Let them cool fully before adding frosting.

Place the softened cream cheese and butter in a bowl and beat until creamy. Gradually add the sifted powdered sugar, beating until light and fluffy. Stir in the vanilla, lemon juice, sour cream, and salt. Beat just until the mixture becomes smooth and easy to spread. If the frosting feels loose, chill it for 15–20 minutes.

Set one cooled cake layer on a serving plate. Spread a generous, even layer of frosting across the top. Add the second layer and cover the top and sides with the remaining frosting. Use an offset spatula to make gentle swirls for a homemade finish. Chill the cake for about 30 minutes before cutting. Store any remaining cake in the refrigerator.