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To make the frosting, combine the soy milk, soy powder, coconut flour, maple syrup, and vanilla in your blender.
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Blend the ingredients for 2 minutes.
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With the machine running, slowly add the oil and lemon juice, alternating between the two until both are fully incorporated.
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Pour the mixture into an airtight container and refrigerate for 6 hours or for up to 1 month.
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If you are using this mixture as a sauce, store the mixture at room temperature for up to 1 week.
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To make the cakes, preheat the oven to 325 F.
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Line 2 standard 12-cup muffin tins with paper liners.
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In your food processor, mix together the flour, baking powder, baking soda, xanthan gum, salt, cinnamon, ginger, and nutmeg.
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Add in the oil, agave nectar, rice milk, and vanilla to the dry ingredients just until a thick batter forms.
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Slowly add the hot water and continue mixing until the batter is smooth.
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Using a plastic spatula, gently fold in the carrots just until they are evenly distributed throughout the batter.
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Pour 1/3 cup batter into each prepared cup, almost filling it.
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Bake the cupcakes on the center rack for 25 minutes, rotating the tins 180 degrees after 14 minutes. The finished cupcakes will bounce back slightly when pressed, and a toothpick inserted in the centre will come out clean.
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Let the cupcakes stand in the tins for 20 minutes, then transfer them to a wire rack and cool completely.
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Using a frosting knife, gently spread 1 tablespoon of vanilla frosting over each cupcake.
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Store the cupcakes in an airtight container in the refrigerator for up to 3 days.