For the apples
2 lb. (about 4 large) sweet apples that hold their shape when cooked (such as Braeburn, Golden Delicious, or Jonagold), peeled, quartered, and cored, each quarter sliced into 3 wedges
1 large lemon, finely grated to yield 1 Tbs. zest (reserve for the cake) and squeezed to yield 1 Tbs. juice
3 oz. (6 Tbs.) unsalted butter, cut into 6 pieces
Pinch table salt
For the topping
Unsalted butter, softened, for the pan
1 cup granulated sugar
1 tsp. ground cinnamon
For the cake
4-1/2 oz. (1 cup) unbleached all-purpose flour
1 tsp. baking powder
1/4 tsp. ground cardamom
1/4 tsp. table salt
3 oz. (6 Tbs.) unsalted butter, cut into 1/2-inch cubes and slightly softened
3/4 cup granulated sugar
1 tsp. pure vanilla extract
1 large egg
2 large egg yolks
1/2 cup whole milk
Prepare the apples
In a large bowl, toss the apples with the lemon juice.
In a heavy-duty 12-inch skillet, cook the butter over medium heat until it has a nutty fragrance and there are brown bits on the bottom of the skillet, about 4 minutes. Immediately add the apples and salt and toss gently with a heatproof spatula until well coated. Cook, uncovered, for 5 minutes, tossing once. Toss again, cover, and cook, tossing every 2 to 3 minutes, until the apples are tender, 6 to 8 minutes.
Uncover and cook, stirring gently, until some of the apples begin to brown and any liquid has evaporated, about 2 minutes more. Set aside until cool enough to handle.
Make the topping
Butter a 9x2-inch round cake pan, line the bottom with parchment, and butter the parchment. Have ready a pastry brush and a small bowl of water.
Put the sugar, cinnamon, and 1/3 cup water in a 2- to 3-quart saucepan and cook over medium heat, stirring, until the sugar dissolves. Stop stirring and, using the pastry brush dipped in the water, wash any sugar crystals from the side of the pan. Continue to cook, without stirring, until the caramel begins to color; then swirl the pan until the caramel turns an even, deep amber, about 3 minutes. Immediately pour the caramel into the prepared cake pan, swirling to evenly coat the bottom. Let cool.
Starting in the center of the pan, arrange the cooled apple slices on the caramel in slightly overlapping, tightly packed concentric circles. Set aside.
Make the cake
Position a rack in the lower third of the oven and heat the oven to 350°F.
In a medium bowl, whisk the flour, baking powder, cardamom, and salt.
In a stand mixer fitted with the paddle attachment, beat the butter on medium speed until smooth and creamy, about 1 minute. Scrape the bowl and beater with a rubber spatula. Add 1/4 cup of the sugar, the reserved lemon zest, and vanilla and beat on medium-high speed until well combined, about 1 minute. Scrape the bowl and beater.
With the mixer on medium speed, slowly sprinkle in the remaining 1/2 cup sugar, taking 20 to 30 seconds to add it. Increase the speed to medium high and beat until pale and creamy, 3 to 4 minutes, stopping once to scrape the bowl and beater.
Add the egg and beat on medium speed until combined, about 1 minute. Add the yolks and beat until incorporated, 1 minute. (It’s OK if the batter looks curdled.)
With the mixer on low speed, alternately add the dry ingredients in three additions and the milk in two additions; scrape the bowl and beater as necessary and mix each addition just until smooth.
Using the rubber spatula, spread the batter evenly over the apples. Tap the pan down on the counter once or twice to settle the batter. Bake until the cake springs back when gently pressed and a toothpick inserted in the center comes out clean, 40 to 45 minutes.
Transfer the cake to a rack and run a small, sharp knife around the edge to release it from the pan. Cool the cake in the pan for 20 minutes.
Holding the pan between your palms, rotate it briskly back and forth on the countertop to release the apples from the bottom. Invert the cake onto a cake plate and slowly remove the pan and the parchment. If any apples have shifted, reposition them. Let the cake cool for at least 1 hour before serving.