Preheat the oven to 350 degrees F. Melt the butter in a small sauce pan over low heat, or in the microwave. Set aside to cool.
Butter a 9-inch springform pan, cut a circle of parchment paper and line the bottom of the pan with it, then butter the parchment and set the pan aside.
Zest the lemon then juice it. Set aside 2 tablespoons of lemon juice for the cake batter. Transfer remaining lemon juice to a small bowl, you'll use it to keep the peeled apples from turning brown. If you don't have much left, juice another lemon to be safe.
Add the flour, baking powder, and salt to a mixing bowl and whisk to combine, then set aside.
Add the sugar to the cooled melted butter and whisk for a minute or two until it's opaque.
Add eggs to the sugar mixture one at a time and whisk until fully incorporated and smooth. Finally, add the yogurt, lemon juice, lemon zest, and vanilla to the egg mixture and whisk to combine. Set wet ingredients aside. See note.
**Before you prep the apples, know that you can peel all of the apples or none of the apples. I like to keep the skin on one of them and peel the other two to make a pretty design on top. You do you.**
First, slice the apple that you aren't going to peel. Cut it into quarters, then cut out the core. Using a small paring knife, cut small slices going lengthwise down the apple without going all the way through to the other side (see photo above.) Coat the apple pieces in the extra lemon juice and set aside.
Peel the next apple, then roll it around in the bowl of lemon juice to coat it and keep it from turning brown. Cut it into quarters and cut out the core. Repeat the process of making narrow slices in the apple going lengthwise without cutting all the way through. Coat it in more lemon juice. Repeat this process with the last apple.
Gently fold the dry ingredients into the wet in 3 batches, making sure all of the dry ingredients are completely incorporated. (This batter is good at hiding flour, so be on the lookout!)
Transfer the batter to the prepared pan and use an offset spatula to smooth it toward the edges and make sure it's level on top. Making the design of your choice on top, gently press the apple quarters core-side down into the batter about ¼-½ centimeter deep, just so a little bit of the batter comes up around the edges. (You may have pieces left over depending on the size of your apples.)
Bake 50-55 minutes until the edges of the cake are deeply golden and starting to come away from the edges. Cool the cake completely in the pan. Dust with powdered sugar just before serving.