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Take one of the apple quarters, and cut it into thin slices but do NOT cut all the way down, just about halfway down through the apple. You want it to be attached at the bottom, you’re basically just scoring the rounded exterior part of the apple quarter (look at the photos in the post for a visual). Repeat with all the apple quarters so they all have score lines cut into the rounded exterior part. Rub the scored part of the apple quarters with the leftover lemon from juicing to get a bit of juice on the exterior (this will help keep them from browning while you make the batter). Set them aside.
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Preheat the oven to 350 degrees Fahrenheit. Grease a roughly 9 and 1/2 by 13-inch rectangle casserole dish and line the bottom with parchment paper and set it aside.
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In a large bowl, mix together the flour, baking powder, baking soda, cinnamon, nutmeg, and salt until combined. Set aside.
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In the bowl of a stand mixer fitted with the paddle attachment, cream together the butter and the sugar at medium low speed until combined and slightly pale in color. Add the olive oil and mix until smooth, then add the eggs one at a time, mixing well after each addition.
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Add the lemon juice, zest, and vanilla extract and mix until combined. Then reduce the speed to low and alternate between adding the flour mixture and the milk in three increments, mixing until the batter is *just* combined (do not overmix).
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Pour the batter into the prepared rectangle baking pan. Evenly space out the apple quarters on the cake, scored side facing up. Place the pan in the oven and bake until a toothpick inserted into the center comes out clean, about 35 minutes.
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While the cake is baking, you can make the syrup. Heat the honey, water, and cinnamon together in a small pot over medium heat. Bring to a boil, then reduce heat to low and simmer for 5 more minutes. Remove from heat and set aside.
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When the cake is finished baking, remove it from the oven and place it on a cooling rack. Evenly drizzle it with the syrup and allow it to cool to room temperature while it absorbs the syrup. Slice and serve and enjoy! The cake will keep in an airtight container in the refrigerator for up to 1 week, it’s best enjoyed warm, so you can take a slice and microwave it for about 30 seconds to warm it up before diving into cake leftovers :)