Heat oven to 350°F. Line an 8-inch square metal baking pan with parchment so it overhangs two sides; lightly brush the parchment and exposed sides of the pan with a few drops of oil.
In a medium bowl, whisk the flour, cinnamon, baking powder, baking soda, and salt until evenly combined. Set aside.
In a large bowl, whisk the egg, oil, applesauce, maple syrup, vanilla, and lemon zest until smooth and slightly emulsified, about 30–45 seconds.
Sprinkle the dry mixture over the wet and fold with a spatula just until no dry streaks remain. Fold in the grated zucchini last, distributing it evenly without overmixing.
Let the batter rest for 10 minutes to hydrate the flour. Scrape into the prepared pan, smooth the top, and tap the pan firmly on the counter 2–3 times to release large air bubbles.
Bake for 34 minutes, or until the center springs back when lightly pressed and a toothpick comes out with a few moist crumbs attached.
Cool the cake in the pan on a rack for 15 minutes. Lift it out using the parchment and cool until just barely warm, then place in the fridge for 20 minutes to finish cooling before frosting.
Make the frosting: In a mixing bowl, beat the cream cheese with Greek yogurt, maple syrup, vanilla, and lemon juice on medium speed for 1 minute, then on high speed for another minute until fluffy and smooth. If it seems loose, chill for 10 minutes to set.
Spread frosting over the cooled cake. Slice into 12 squares and serve.