1) To begin the frosting: Put the cashews in a large bowl and add enough water to cover them by 1 inch. Let them stand until softened, at least 2 and up to 4 hours. Drain well.
2) To make the cake: Put 1 cup of the shredded carrots in a medium saucepan. Add enough cold water to barely cover the carrots and bring to a boil over high heat. Reduce the heat to medium low and simmer until the carrots are tender, about 10 minutes. Drain well. Purée the carrots in a mini food processor or blender. You should have 1/3 cup purée.
3) Position a rack in the center of the oven and preheat the oven to 350° F. Lightly and evenly coat the inside of an 8-inch square baking pan with solid coconut oil. Dust with flour and tap out the excess flour.
4) Beat the brown sugar and coconut oil with an electric hand mixer at high speed until slightly paler in color, about 1 minute. Add the coconut milk, vanilla, and the carrot purée and beat until foamy, about 1 minute more. Whisk the all-purpose and whole wheat flours, cinnamon, baking powder, and baking soda, allspice, and salt together in a large bowl, being sure to break up any baking soda clumps. Pour in the brown sugar mixture and mix just until combined into a thick batter. Stir in the remaining 1 cup shredded carrots with the walnuts and orange zest. Spread the batter evenly in the cake pan.
5) Bake until the cake springs back when pressed gently in the center with a fingertip, about 35 minutes. Let cool completely in the pan on a wire cake rack.
6) To complete the frosting: Transfer the drained cashews to a food processor. Add the confectioners' sugar, coconut oil, vanilla, lemon juice, and salt, and pulse until the nuts are finely chopped. With the machine running, and occasionally stopping to scrape down the sides of the work bowl, add enough of the coconut milk to make a smooth, spreadable frosting. Spread the frosting over the top of the cake. Cut into squares and serve.
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