Preheat the oven to 350°F. Butter and flour a 9-inch springform pan. Line the bottom with parchment paper.
Toast the pistachios for 8 minutes until the pistachios are just beginning to brown. Remove them from the oven and allow them to cool. Roughly chop 1/2 cup (80 grams) of the toasted nuts
Place the flour, baking powder and salt in a bowl and whisk for 30 seconds to combine the dry ingredients. Add the chopped pistachio nuts and whisk to combine.
Peel and chop two of the apples into 1/2-inch dice. You should have about 3 cups or 300 grams of chopped apples. For the third apple, peel it and then cut into quarters. Cut each quarter into 6 slices for a total of 24 slices. These slices will be used to decorate the top of the cake.
Place the white sugar and brown sugar in the bowl of a stand mixer. Zest the lemon directly onto the sugar. Use your fingers to rub the lemon zest into the sugar and break up any clumps in the brown sugar. Beat the sugar and butter on medium speed for 4-5 minutes until it looks fluffy and is lighter in color.
Add the eggs one at a time, beating on medium speed until the egg is incorporated. Scrape down the sides and bottom of the bowl.
Add half the flour-nut mixture and mix on low speed until the flour mixture is incorporated. Add all the buttermilk and mix on medium-low speed to incorporate. The batter will look a little curdled but that’s okay. Add the remaining flour and mix on low speed until incorporated. Any flour left on the sides of the bowl will be mixed in by hand.
Scrape the sides and bottom of the bowl. Add the chopped apples all at once. Use a silicone spatula to mix in the chopped apples into the cake batter. Scrape the batter into the prepared springform pan and use an offset spatula to smooth the batter. Arrange the sliced apples on top.
Bake cake in the oven 60-70 minutes until the center of the cake registers 200°F or a cake skewer comes out clean or with just a few crumbs clinging to it.
Remove the cake from the oven. Let it cool for 10 minutes in the springform pan and then remove the sides. While the cake is still warm, warm the apricot jam in a small saucepan until it becomes running. Brush the top of the cake with the apricot jam and sprinkle with the remaining pistachio nuts.
All the cake to cool completely and then serve.