Prepare my pie crust recipe through step 5. In a large bowl using a silicone spatula or wooden spoon, stir the apples, lemon juice, flour, all of the spices, sugar, and vanilla extract together until thoroughly combined. Set filling aside as the oven preheats; this time allows the apples to begin letting off their juice. Preheat oven to 400°F (204°C). On a floured work surface, roll out one of the discs of chilled dough (keep the other one in the refrigerator). Turn the dough about a quarter turn after every few rolls until you have a circle 12 inches in diameter. Carefully place the dough into a 9-inch pie dish. Tuck it in with your fingers, making sure it is smooth. Spoon the filling into the crust and discard some of the leftover juices in the bottom of the bowl. Dot the pieces of butter on top of the filling. Remove the other disc of chilled pie dough from the refrigerator. Roll the dough into a circle that is 12 inches diameter. Carefully lay the dough over the filling. Use a small paring knife or kitchen shears to trim off excess dough from the sides. Cut slits in the top to form steam vents. Crimp or flute the pie crust edges to seal. Lightly brush the top of the pie crust with the egg/milk mixture. Sprinkle the top with coarse sugar. Place the pie onto a large baking sheet and bake for 20 minutes. Keeping the pie in the oven, turn the temperature down to 375°F (190°C) and bake for an additional 30–35 minutes. After the first 20 minutes of bake time, I place a pie crust shield on top of the pie to prevent the edges from browning too quickly. Allow the pie to cool for 3 full hours at room temperature before serving. This time allows the filling to thicken up. Cover pie leftovers tightly and store in the refrigerator for up to 5 days.