Preheat your oven to 350°F (175°C). Grease a 10-inch round cake pan with butter and dust it with flour. In a medium saucepan, bring the maple syrup to a boil over medium-high heat. Reduce the heat to maintain a steady simmer and cook for about 40 minutes, until the syrup darkens and thickens, reducing to about 1 cup. Watch it closely to prevent it from boiling over or burning.
Remove the syrup from the heat and whisk in 2 tablespoons of butter until melted. Carefully pour the hot syrup mixture into the prepared cake pan.
Peel, core, and halve the apples. Cut each half into 8 thin slices. Arrange the apple slices in tight, concentric circles over the syrup in the bottom of the pan. You can create a second layer if needed.
In a small bowl, whisk together the flour, baking powder, salt, and baking soda. In a separate medium bowl, whisk together the eggs, egg yolks, buttermilk, and vanilla.
In the bowl of a stand mixer or in a large bowl with a handheld mixer, beat the remaining 1 cup of butter and the granulated sugar on medium speed for 6-7 minutes, until pale and fluffy. Scrape down the bowl as needed.
With the mixer on low, add one-third of the flour mixture, followed by half of the egg mixture, beating just until combined after each addition. Add another third of the flour mixture, followed by the remaining egg mixture. Finally, fold in the last of the flour mixture with a spatula until just incorporated.
Spoon the batter in dollops over the apples in the pan. Gently spread the batter into an even layer with an offset spatula or the back of a spoon.
Bake for 60 to 70 minutes, or until the cake is golden brown and a tester inserted into the center comes out clean. Let the cake cool in the pan on a wire rack for 1 hour.
While the cake is still warm, run a knife around the edge to loosen it. Place a large serving plate over the pan and carefully invert the cake. If any apple slices stick to the pan, gently lift them out and place them back on the cake. Serve warm.