For the Cake:

1. Preheat oven to 375°F. Cover one sheet pan with parchment paper or a silicone mat.
2. In a large bowl, combine the dry ingredients (flour, baking soda, baking powder, salt, and sugar). Whisk very well.
3. In a separate bowl, whisk together the milk, applesauce, vegetable oil, water, vinegar, vanilla, and lime zest.
4. Pour the wet mixture into the dry mixture, and stir to combine until a thick batter forms.
5. Using a small cookie scoop, scoop out and drop 16 small mounds of the batter on the prepared cookie tray, leaving about an inch of space between each mound.
6. Bake for about 6-8 minutes, until puffy and lightly golden along the edge. Remove from the oven and let sit for 10 minutes before transferring the cakes to a cooling rack to cool completely.

For the Icing and Assembly:

1. To the prepared frosting, stir in the key lime juice and zest until completely incorporated.
2. Using a spatula or piping bag fitted with a round tip, cover the bottoms of 8 cooled cake rounds with a thick layer of frosting. Top each frosted half with the remaining 8 plain cake rounds. If using, garnish the edges of the frosting with the ground graham crackers.
3. Serve immediately, or store in the refrigerator for up to a week. When ready to serve, leave out at room temperature for 10 minutes to soften the frosting slightly.