FOR CAKE:
1. Preheat the oven to 350°F and lightly grease an 8-inch or 9-inch round or square pan that’s at least 2 inches deep. Line the pan with parchment or a reusable pan liner. 

2. In a medium bowl, whisk together flour, baking powder, salt, and baking soda. Set aside. 

3. In a separate bowl or large measuring cup, whisk together water, sugar, vegetable oil, lemon juice, organic lemon emulsion, and vanilla. Stir to dissolve the sugar. 

4. Add the wet ingredients to the dry and stir to combine. It's OK if a few small lumps remain. 

5. Pour the batter into the prepared pan. 

6. Bake the cake for 30 to 35 minutes until the top feels set, the edges begin to pull away from the sides of the pan, and a toothpick inserted into the center comes out clean. Once the cake is cool enough to handle, turn out onto a wire rack to cool completely. 

FOR FROSTING:
1. In a medium bowl, beat together all of the frosting ingredients, starting with the lesser amount of water. Mix until the frosting is smooth and silky. Adjust with additional water as needed. Spread over the cooled cake.  

2. Garnish the cake with candied lemon** or fresh berries, if desired. 

Store the vegan lemon cake, covered, at room temperature for several days. Freeze for longer storage. 

**FOR CANDIED LEMON TOPPING:
1. Slice 1 1/2 lemons into very thin rings (if you have a mandoline, now is the time to use it). Tease out any seeds. Juice the remaining 1/2 lemon and set aside.

2. Bring a small pot of water to a boil, add the lemon slices, and cook for 1 minute. Then use a slotted spoon to transfer them to an ice bath.

3. Bring 1 cup (198g) sugar, 1 cup (227g) water, and the reserved lemon juice to a boil, stirring to dissolve the sugar. Lower to a bare simmer, add the lemon slices, and simmer very gently for 45 minutes to 1 hour, stirring occasionally, until the slices are translucent, with no white bits remaining. Use a slotted spoon to transfer to a parchment-lined baking sheet to cool.

For best results, store in an airtight container and use within 1 day. You can save the lemon jelly left in the pot for spreading over toast or mixing into yogurt.