1. Position a rack in the middle of the oven and preheat the oven to 350°F.
2. Trim the ends from the lemons and cut the fruit into slices ¹/8 inch to ¹/4 inch thick with a very sharp knife or a mandoline; discard any seeds.
3. Melt 1/4 cup of the butter with the brown sugar in an ovenproof, nonstick 10-inch skillet over medium heat. Add the lemon slices, increase the heat to high, and boil, stirring, for 1 minute. Remove the skillet from the heat and arrange the slices in an orderly pattern in the bottom of the skillet.
4. Sift the flour, baking powder, and salt together into a medium bowl.
5. Beat the remaining ¹/2 cup of butter with an electric mixer on medium speed in a large bowl until light and fluffy. Add the granulated sugar and zest and beat, scraping down the side of the bowl until light and fluffy. Add the egg yolks and vanilla and beat just until blended. Reduce the speed to low and add the flour mixture alternately with the milk, scraping down the side of the bowl after each addition and beating just until blended.
6. Beat the egg whites with clean beaters on medium speed in a large clean bowl just until foamy. Increase the speed to medium-high, add the cream of tartar, and beat just until the egg whites form stiff peaks. Add one quarter of the whites to the batter and fold in using a whisk or a rubber spatula; continue to gently fold in the whites, one quarter at a time, being careful not to overmix.
7. Pour the batter over the lemon slices and gently smooth the top with a rubber spatula. Bake for 45 to 50 minutes, until the top is golden brown and a wooden pick inserted in the center comes out clean. Let cool in the pan on a wire rack for 10 minutes.
8. Loosen the edge of the cake with a rubber spatula all around, invert it onto a heat-proof serving platter, and leave the pan over the cake for 5 minutes. Remove the pan and serve the cake warm, cut into wedges.