Preheat the oven to 350 degrees F. Lightly grease a 9 x 13-inch pan.
In a large mixing bowl, whisk together the sugar, oil, lemon juice, lemon zest, applesauce, and vanilla until well combined. Add water and mix thoroughly.
Add the flour, baking soda, and salt, and stir until just combined. (Your batter may have a few small lumps and that's okay. You don't want to overmix.)
Fold in the poppy seeds.
Pour the batter into the pan. Bake for 35 minutes or until the edges are golden and begin to pull away from the pan. Take a toothpick, insert it into the middle of the cake and make sure it comes out clean. If it doesn’t bake in 5 minute increments until it does.
In the meantime, prepare your glaze by simply whisking together the powdered sugar and lemon juice. Start with 4 teaspoons, and if it’s too thick, add another one.
Allow the cake to cool completely (or at least 15 minutes if you just can't wait!) before drizzling with glaze. Enjoy!