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Preheat oven to 350 degrees F (180 C) and line 2 (8-inch) cake pans with parchment paper. Pour aquafaba into bowl of stand mixer and beat until its begins to look light and fluffy (about 3 minutes). Slowly drizzle in cane sugar while beating on high and continue until it forms stiff peaks (it should take about 5 more minutes).
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In separate large mixing bowl, stir together the brown sugar, lemon juice, coffee, melted coconut oil, oil, and extract. Add the flour, espresso powder, salt, and baking powder and gently stir.
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Remove the mixer bowl from the mixer and use a large spatula to gently fold in the aquafaba into the other bowl of ingredients just until fairly combined. (It's actually better to leave some small lumps then to overmix and lose volume.)
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Divide batter into the 2 pans as evenly as possible. Bake for 30-40 minutes until toothpick inserted in center comes out clean and edges are lightly golden. Remove the cakes from oven and cool completely before frosting. (I prefer to cover them and place them in the freezer for at least an hour.)