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Position a rack in the center of the oven and preheat to 350ºF. Line an 8-inch square baking pan with 2 crisscrossing pieces of parchment paper, leaving an overhang on each side. (Alternatively, grease the pan with 1 teaspoon softened unsalted butter.)
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Make the cake: In a large bowl, stir together the grated carrots, eggs, sugar, and oil until combined. Place a medium-mesh strainer over the bowl and add the oat flour, sweet rice flour, millet flour, tapioca flour, baking powder, baking soda, salt, cinnamon, and ginger to the strainer. Sift the dry ingredients into the wet ingredients, stirring until combined. Stir in the walnuts.
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Scrape the batter into the prepared pan and smooth into an even layer. Bake until a toothpick inserted into the center comes out clean, 25 to 35 minutes. Remove from the oven and let cool 20 minutes in the pan. Turn the cake out onto a serving board to cool completely.
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Make the frosting: In the bowl of a stand mixer fitted with the paddle attachment, combine the cream cheese, butter, and powdered sugar and mix on medium speed until smooth, 2 to 3 minutes. Add the maple syrup, lemon juice, and salt and mix on medium-high speed until light and fluffy, 2 to 3 minutes longer.
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Spread the cream cheese frosting over the cooled cake and sprinkle with toasted walnuts and a drizzle of maple syrup. Cut into squares and serve.