Preheat your oven to 350 degrees Fahrenheit. Grease or spray the bottom of a 9-inch can pan with non-stick cooking spray then line with parchment paper. You can also use a springform pan or if you don't have a round pan, an 8 or 9-inch square pan.
In a large mixing bowl, use a hand mixer to combine the sugar and olive oil until homogenous and fluffy. If you don't have a hand-mixer, you can use a fork or metal whisk to vigorously mix to combine.
Whisk in the applesauce and vanilla until the ingredients are fully incorporated into the sugar mixture.
Gently fold in the flour, salt, and baking powder until there are no dry clumps of flour remaining in the batter. Once the dry ingredients are incorporated, stop mixing the batter.
Pour the mixture into the prepared cake pan and bake for 35 minutes, or until a toothpick can be cleanly removed from the center of the cake. Let the cake cool for 10 minutes in the pan then invert the cake into your hand or onto a wire cooling rack, and remove the parchment paper. When you're ready, transfer the cake to a serving platter or stand.
Prepare the lemon glaze by whisking together the lemon juice and powdered sugar until smooth and runny. Spoon the glaze over the cake, serve, and enjoy!