Preheat oven to 350°F (175°C). Grease two 9-inch round cake pans, line bottoms with parchment circles, and lightly spray again. Set aside.
Wash, peel, and finely grate 4-5 medium carrots to yield exactly 3 cups (360g). Use a box grater or food processor for fine shreds.
If using nuts, spread 1 cup chopped pecans or walnuts on a baking sheet and toast at 350°F for 6-8 minutes until fragrant and golden. Cool completely.
In a large bowl, whisk together flour, baking soda, cinnamon, ginger, nutmeg, and salt for at least 30 seconds to distribute evenly.
In another large bowl, whisk brown sugar, granulated sugar, oil, and applesauce until smooth. Add eggs one at a time, whisking well after each, then stir in vanilla.
Pour wet ingredients into dry and gently fold with a spatula until just combined—do not overmix. A few small lumps are okay.
Gently fold in the grated carrots and toasted nuts (if using) until evenly distributed with no dry flour pockets.
Divide batter evenly between the prepared pans and smooth tops. Bake for 30-35 minutes until a toothpick comes out clean or with a few moist crumbs.
Cool cakes in pans for 15 minutes on wire racks. Run a knife around edges, invert onto racks, remove parchment, and cool completely (at least 1 hour) before frosting.
Beat softened cream cheese and butter until smooth and fluffy (2-3 minutes). Gradually add sifted confectioners' sugar, then vanilla and pinch of salt. Beat until creamy. Add 1 tbsp cream if too thick.
Place one cooled layer on a serving plate. Spread 1-1½ cups frosting on top. Add second layer, apply a thin crumb coat, refrigerate 15-20 minutes, then frost completely. Decorate with pecans if desired.