Peel, core, and slice the apples ¼" thick. Try to slice them as close to the same size as possible so they cook evenly.
In a large stockpot melt the butter over medium heat. Whisk in the cornstarch, lemon juice, both sugars, salt, and cinnamon. Add in the apples and cook and stir until the apples are evenly coated and starting to soften. Remove from heat and allow to cool slightly.
Remove the pie from the fridge. Brush the inside of the bottom pie crust all over with the beaten egg. Spoon the apple filling into the pie dish and cover with the second rolled out portion of dough.
If topping with a lattice: Use a pastry wheel or a pizza knife to cut the dough into 1" strips. Arrange them going one way over the top, then fold back every other strip and add another going in the opposite direction. Repeat this, alternating the strips as you go, until the entire pie is covered. See post for visual, or here is a video tutorial: How to Lattice Pie Crust:
If using a solid topping: Cover the filling and cut a few slits in the top for steam to escape. Brush the top of the pie with beaten egg and sprinkle with coarse sugar, if desired.
Keep the pie in the fridge and preheat your oven to 425°.
Place the pie on a baking sheet large enough to catch any drips that may bubble over. Bake for 15 minutes and then, without taking the pie out of the oven, lower the temperature to 350°. Continue baking for 40-45 minutes, until the filling is cooked through and the apples are soft.
Allow the pie to cool slightly before serving. Serve warm with a side of vanilla ice cream.