Preheat the oven to 350 degrees and grease an 8 or 9 inch round cake pan that is at least 3 inches deep. Line the bottom with a circle of parchment paper, and sprinkle a little sugar all around the edges of the cake pan. Tip out any excess that does not stick to the buttery sides of the pan.
In a large bowl or a stand mixer, beat together the sugar and vegan butter really well for 3 to 4 minutes.
In another bowl, combine the aquafaba, lemon zest and juice, applesauce, salt, vanilla and nondairy milk, and whisk them together well.
Sieve the flour and baking powder into the butter mixture. Stir together gently by had or on the lowest mixer speed, and then gradually pour in the milk mixture, a few tablespoons at a time, mixing gently as you go, until a thick batter is formed. Do not overmix. Be gentle and keep going only until all the dry flour has been combined. A few small lumps are fine.
Spoon into the prepared cake pan and bake in the middle of the oven for 45 to 50 minutes. the cake should be coming away from the edges of the pan, and if you press gently in the middle with a finger, it should spring slowly back up. A toothpick inserted right into the middle of the cake should come out clean. Turn out, very gently, onto a cooling rack and peel off the parchment paper. If it is a little around the edges because the sugar has baked on, just gently run a knife around the edges first.