Line a shallow bowl with parchment paper. Set aside.

Cut the butter into chunks. Place the butter in a medium saucepan and cook over medium-low heat, stirring frequently, until the butter gets foamy and bubbly. Continue stirring and cook for 5-8 minutes, until the butter smells nutty and has brown bits on the bottom. Keep a close eye on it so the butter doesn't burn!

Once browned, immediately pour the butter into the parchment-lined bowl. Place in the freezer for 60 minutes, stirring every once in a while, until very firm and cold. Alternatively, firm the butter in the fridge overnight.

Use the parchment paper to remove the butter from the bowl. Discard the parchment and cut the cold brown butter into 1-inch chunks.

****Combine the flour, sugar, and salt in the bowl of a food processor. Toss the flour with the butter, and pulse a few times until you have both small pea-sized butter chunks and larger almond-sized chunks.

Pour the flour-butter mixture into a large bowl. Combine the apple cider vinegar and ice water in a measuring cup, and drizzle it over the flour mixture. Run your fingers through the flour to incorporate the ingredients. Add more ice water 1/2 Tablespoon at a time if necessary. Then, quickly knead the dough until it comes together.

Divide the dough into 2 pieces. Flatten both pieces into 1-inch thick discs and wrap them with beeswax wrap or plastic wrap. Refrigerate for at least 1 hour or up to 5 days. You can also freeze the dough for up to 2 months.

Start by peeling and coring the apples. Cut the apples into 1/8-1/4'' slices. Keep them as uniform as possible to ensure even baking.

Place the apples in a large bowl. Drizzle in the lemon juice and toss to combine. Then, add the brown sugar, all-purpose flour, cinnamon, apple pie spice, cardamom, and salt. Mix well and let the apples sit for a couple of minutes until there are no dry spots.

Preheat the oven to 400°F. Place your oven racks on the second and third rack of your oven (from the bottom). Microwave the caramel sauce for about 30 seconds, or until it is a drizzling texture.

On a floured surface, roll one of the pie dough discs into a 12'' round. Wrap the dough over your rolling pin and unroll it into a deep 9'' pie dish, letting the extra dough hang over the sides. Gently press the dough into the bottom and sides of the pie dish, making sure not to tear it. (If you do, just press the dough to fill in the cracks). Place your pie dish into the fridge. Next, roll out your next pie dough disc into a 12'' round.

Fill the bottom pie shell with the apple filling, mounding it a bit in the center. Drizzle the caramel sauce evenly over the apple pie filling. Then, using your rolling pin again, roll and unroll the second 12'' dough round over the top of the pie. Press the edges of the bottom and top crusts together to form a seal. Trim with a knife or shears until you have about a 1/2'' left.

Using your fingers, crimp the edges of the pie crust. Use any leftover pie dough to make decorations, as desired. I like to roll out the leftover dough, cut out shapes with leaf-shaped cookie cutters, and press them on top of the pie!

Place the pie in the fridge for about 20 minutes. Meanwhile, whisk together the egg and 1 Tablespoon of water in a small bowl.

Once chilled, brush the top of the pie with the egg wash. Sprinkle liberally with Demerara sugar.

Place the pie on the third rack of the oven from the bottom. Place a sheet pan on the rack underneath to catch any drips. Bake 45-50 minutes until the crust is a deep golden brown and the filling is bubbly.

Let the pie cool at room temperature for at least 3 hours. Slice, re-warm if desired, and serve with vanilla ice cream, caramel sauce, and chopped pecans.