Preheat the oven to 350°F. Line a 9-inch cake pan with parchment paper.
To make the batter: In a large mixing bowl, add the olive oil, coconut milk, lemon juice, cane sugar, and eggs. Whisk to combine. (You can also do this with a stand mixer or hand mixer if you prefer).
Add the cake flour, baking powder, sea salt, and lemon zest in that order. Fold together with a spatula until combined into a smooth cake batter. Transfer to the prepared cake pan and spread evenly.
Bake at 350°F for 50 to 60 minutes or until a toothpick inserted in the center comes out clean. Allow the cake to cool completely in the pan.
To make the frosting: In the bowl of a stand mixer fitted with the paddle attachment (or in a bowl with a hand mixer, or whisk), add the butter, powdered sugar, coconut milk, and lemon flavor. Mix on low until combined and then speed mixer up to high and mix for 1 minute or until light and fluffy.
Remove the cake from the pan and spread the frosting on top. Top with candied lemons (optional).
Serve immediately, or store in the fridge for up to 3 days.