Wash and peel the apples well. Slice them into 1/2-inch thick pieces and place them in a bowl with the sugar and lemon juice.

In a large skillet or wide saucepan, melt 4 tablespoons butter. Add the apples and cook over high heat for 15 minutes. Sift in the flour, add the cinnamon and salt, and cook for 2 more minutes. Let the mixture cool completely. You can refrigerate the cooked apples for up to 2 days if needed. I let them sit overnight and made the pie the next day.

For the crust: In the bowl of a mixer, combine the flour, sugar, apple cider vinegar, salt, and 4 tablespoons of the cold butter. Mix briefly, then add the remaining butter in small pieces. Add 1/4 cup ice water and mix until a smooth dough forms.

Assembly: Generously butter a 9-inch (23 cm) pie pan.

Roll out one portion of the dough to fit the pan, including the sides, using a rolling pin. For easier rolling, you can place plastic wrap on top of the dough.

Add the apple mixture and dot with 2 tablespoons cold butter, spreading it evenly. Roll out the second portion of dough the same way and place it on top, sealing the edges well. Cut slits in the top crust, brush with beaten egg, and sprinkle with brown sugar for the topping.

Preheat the oven to 425°F (220°C) with a baking sheet inside for at least 30 minutes. Once preheated, place the pie on the hot baking sheet. Lower the oven temperature to 375°F (190°C) and bake for 1 hour to 1 hour 10 minutes. For best results, wrap the edges of the pie with aluminum foil.