Position a rack in the center of your oven and preheat the oven to 350°F. Butter a 9-inch round cake pan and line it with parchment paper.

In a medium saucepan over low heat, melt the butter. Remove the pan from the heat and add ½ cup of the sugar and 2 tablespoons of water. Whisk the mixture until the sugar dissolves and the mixture is glossy and thickened a little. Pour the melted butter and sugar into the prepared cake pan and smooth into an even layer to cover the bottom. Place the cake pan in the freezer until the butter and sugar layer freezes; leave the pan in the freezer until you are ready to transfer the batter to the pan.

In a small bowl, gently toss the lemon slices with the remaining 2 tablespoons of sugar. Leave the lemon slices in the sugar to macerate for 10 minutes.

Sift the flour, baking powder, baking soda, and salt into a large bowl. Set aside. Combine the eggs with the olive oil in another large bowl and beat vigorously with a whisk until well combined. Add the yogurt and sugar. Make a well in the flour mixture and add the wet mixture in three additions, folding in each addition until no dry bits of flour remain. Set aside.

Fan the lemon slices out to cover the melted butter and sugar in the cake pan, overlapping the slices by as much as 1⁄8 inch. Pour the cake batter directly on top of the lemon slices.

Bake for about 1 hour, until a cake tester inserted into the center comes out with only a few moist crumbs clinging to it. Transfer to a wire cooling rack and let cool for 15 minutes. Invert the cake onto a serving plate and carefully remove the parchment-paper lining. Allow the cake to cool for 1 hour more before serving