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Preheat the oven to 400°F. Spray a 9-inch round cake pan with non-stick spray and line the bottom with parchment. Dust the sides with flour and tap out the excess.
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In a small bowl, stir together the almond milk and lime juice. In a medium bowl, combine the lime zest, whole wheat flour, baking powder, baking soda, and salt.
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Combine the coconut oil and 2/3 cup of sugar in a large bowl and beat with an electric mixer until well combined. Beat in the vanilla and the egg.
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With the mixer on low speed, alternately beat in the flour and almond milk mixtures in three additions, beginning and ending with the flour.
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Spoon the batter into the prepared pan and spread it evenly. Scatter the blackberries over the top and sprinkle with the remaining tablespoon of sugar.
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Bake for 30 minutes, until a toothpick inserted in the center comes out clean. Cool the cake in the pan for 15 minutes, then turn out and cool completely on a rack.