Preheat oven to 300°F. Spread nuts on a parchment-lined baking sheet and toast for 6–8 minutes until fragrant. Cool completely.
Increase oven temperature to 350°F. Grease and line two or three 9-inch cake pans with parchment paper.
In a large bowl, whisk brown sugar, granulated sugar, oil, eggs, applesauce, and vanilla until smooth.
In another bowl, whisk flour, baking powder, baking soda, salt, cinnamon, ginger, nutmeg, and cloves.
Fold dry ingredients into wet ingredients until just combined. Do not overmix.
Fold in grated carrots and half of the toasted nuts.
Divide batter evenly into prepared pans. Bake 25–35 minutes until the center springs back and a toothpick comes out with moist crumbs.
Cool cakes in pans for 1 hour, then remove and cool completely on a rack.
Beat cream cheese and butter until smooth. Gradually add confectioners’ sugar, then mix in vanilla and salt until fluffy.
Level cake layers if needed. Spread frosting between layers and over top and sides. Chill at least 20 minutes before slicing.