Preheat the oven to 350°F (175°C). Line a 9-inch round cake pan (or 8x8 square) with parchment and lightly grease the sides.
In a small bowl, stir the non-dairy milk and vinegar. Let it sit 5 minutes to make a quick vegan “buttermilk”.
In a large bowl, whisk flour, brown sugar, baking powder, baking soda, salt, cinnamon, nutmeg, and ginger until no lumps remain.
Whisk the buttermilk, oil, maple syrup, vanilla, and applesauce together.
Pour into the dry ingredients. Mix with a spatula until just combined. Do not overmix.
Fold in grated carrots and any add-ins. The batter will be thick—this is right.
Transfer to the pan and smooth the top.
Tap the pan gently on the counter to release air bubbles.
Bake for 35–42 minutes, until the center is set and a toothpick comes out with a few moist crumbs.
Cool in the pan 15 minutes, then lift out to a rack to cool completely before frosting.
Make the frosting: Beat vegan butter until smooth. Add vegan cream cheese and beat until just combined. Gradually add powdered sugar, vanilla, and salt.
Beat until creamy and spreadable. Chill 15 minutes if too soft.
Frost the cooled cake. Sprinkle with chopped nuts if you like. Slice and enjoy.