Preheat oven to 350°.
In a large bowl, whisk the syrup, applesauce, milk, vinegar, and vanilla together. In a separate bowl, combine the flour, cane sugar, brown sugar, baking powder, baking soda, salt, and spices together.
Pour the dry ingredients into the wet ingredients, then use an electric mixer to stir together. Gently fold in the carrots and walnuts with a spatula until just combined.
Line two 8" cake pans with parchment rounds (or grease really well with coconut oil), then divide the batter evenly into the pans. Bake the cakes for 40-45 minutes, or until a toothpick comes out clean. Remove them from the oven, then wait 10 minutes before running a knife along the edges and carefully inverting them onto a cooling rack. Wait until the cakes cool completely before icing them.
For the icing: Whisk the butter, milk, syrup, and vanilla together, then add in the powdered sugar and cinnamon and whisk until the icing is smooth.
To assemble: Place a large dollop of icing on top of the one cake, then use an offset spatula or the back of a spoon to smooth it out. Place the second cake on top, and top it with the remaining icing. Slice and enjoy!