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Butter a 9-inch round springform pan and line the bottom with parchment paper. I do this by trapping the parchment between the bottom and the sides of the pan, it makes for easy movement later on. Preheat the oven to 350°F.
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In the bowl of a stand mixer fitted with the paddle attachment, combine butter, white sugar and 1/4 cup brown sugar. Beat at medium speed until light and creamy. Scrape down the sides of the bowl once or twice during the mixing.
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Crack in the egg and add the vanilla. Beat again until very well combined. Scrape down the sides of the mixing bowl.
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Sift the flour and baking soda into a separate bowl. Whisk in the salt and nutmeg. Measure out 2 tablespoons of this flour mixture and toss with the rhubarb and carrots in another bowl.
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With the mixer on Low, add the dry ingredients to the creamed butter in three parts, alternating the buttermilk with the flour, finishing with the flour. Mix well with each addition.
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Add the rhubarb & carrots to the cake batter and gently stir to combine. Scrape the batter into the prepared cake pan and smooth the top.
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Mix together the remaining 1/4 cup of brown sugar with the cinnamon. Sprinkle over the cake batter. Run a butter knife through the cake, making a few figure eights to swirl the cinnamon into the batter.
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Place the cake on a middle rack in the centre of the oven and bake until a toothpick inserted in the middle comes out clean. About 1 hour or 1 hour and 10 minutes. It will get quite dark on the top.
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Cool the cake for 10 minutes on a wire rack, then loosen around the edges and remove the ring. When cool, frost generously with Whipped Mascarpone Frosting (recipe below), then slice and serve.