Line a 9-inch square baking pan with two strips of parchment paper so that the paper overhangs by a few inches all around. This will make the cake easy to remove for slicing later. You can skip the paper step if you want to cut and serve the cake right out of the pan. You can also use a 9-inch springform pan if you don't have a square one that size.
Brush the lined pan or the unlined springform pan with some of the melted butter to coat all sides.
Beat the eggs and sugar in the bowl of a standing mixer on medium-high speed, until the eggs become thick, pale and creamy. Combine the flour, baking powder, baking soda, salt and lemon zest in a bowl.
Lower the mixer speed to stir in the flour mixture in two additions, alternating with the milk. Fold the diced apple into the batter.
Scrape the batter into the pan. Arrange the sliced apples on top. Drizzle the remaining butter over the top of the cake and sprinkle evenly with the pine nuts and coarse sugar.
Bake about 40-45 minutes, until the cake is golden brown and the center is firm when touched with a fingertip. Cool slightly before removing from the pan. Serve warm or at room temperature.