Start by making the crust. Grate 4 tablespoons of chilled butter with grater and place in the freezer. Cut the remaining 16 tablespoons of butter into small cubes.
In a food processor, pulse the flour, sugar, and salt until combined. Add the cubed butter and pulse until combined. The mixture should look crumbly with the butter distributed as small 'pebbles' throughout the flour. Add the grated butter and toss the mixture with a fork until the grated butter pieces are separated and coated with flour.
Sprinkle 1/4 cup of ice water over the mixture and pulse just until combined. Sprinkle the remaining 1/4 cup of ice water over the mixture and pulse until combined.
Transfer the dough it to a floured work surface and gently press the dough until the flour is fully incorporated. Divide the dough in half, flatten each into a 1 inch thick disk, and wrap tightly in plastic wrap. Chill in the refrigerator for at least 2 hours.
To make the filling, start by thinly slicing your apples. In a large bowl, gently toss the apples and lemon juice until the apples are coated.
In a small mixing bowl, combine the flour, nutmeg, cinnamon, and salt. Sprinkle over the apples and gently stir to combine. Set aside.
In a small saucepan, melt butter over medium-high heat. And the brown sugar and milk, whisk to combine. Bring the mixture to a simmer and reduce heat to medium-low. Keep at a low simmer for about 10 minutes, stirring regularly. If the mixture is getting too thick, add 1 tablespoon of milk at a time until the consistency is like a pourable caramel sauce. Remove from heat and add the vanilla.
Assembling the Pie
Preheat oven to 375 degrees F.
Remove your pie dough from the fridge. Roll out one of the disks of chilled dough on a floured work surface until it is a circle about 12 inches in diameter. Carefully place the dough into a 9x2 inch pie dish.
Fill the pie crust with the apples. Drizzle with 1/2 cup of the caramel sauce, reserving the rest for topping.
Remove the other disk of pie dough from the refrigerator. To create a lattice top crust: Roll the dough out into another 12 inch circle. Using a pizza cutter and a clean ruler, cut 10 strips 1 inch wide. Carefully thread the strips over and under one another, pulling back strips as necessary to weave.
Using a sharp knife, trim the extra overhang. Press down the edges of the dough with a fork or your fingers. Brush the lattice crust with a very thin layer of the beaten egg and sprinkle with a little cinnamon and coarse sugar.
Place the pie on a large baking sheet and bake for 20 minutes. Cover the edges of the crust with aluminum foil and bake for an additional 40-50 minutes. The pie will be done when the caramel begins to bubble up.
Allow the pie to cool for 4 hours on the counter before serving. Drizzle each serving with the extra caramel sauce. It is best served on the same day, but it can be covered tightly and stored in the refrigerator for up to 3 days.