Prepare the Pie Dough
In a large bowl, whisk together the flour, sugar, and salt. Add the cold, cubed butter and use a pastry cutter or your fingers to work it into the flour mixture until you have pea-sized pieces. Drizzle in ice water, 1 tablespoon at a time, gently mixing until the dough just comes together. You might not need all the water.Chill the Dough
Divide the dough in half, form into discs, wrap in plastic, and refrigerate for at least 1 hour or up to 3 days.Brown the Butter and Infuse with Sage
In a medium saucepan, melt the butter over medium heat. Continue cooking, swirling occasionally, until the butter turns amber and smells nutty, about 5-7 minutes. Remove from heat, add the chopped sage leaves, and let steep for 10 minutes.Prepare the Apple Filling
In a large bowl, toss the sliced apples with lemon juice. In a separate bowl, combine the sugar, flour, cornstarch, cinnamon, nutmeg, allspice, and salt. Sprinkle this mixture over the apples and toss to coat evenly. Strain the brown butter, reserving the sage bits, and pour over the apples. Add the vanilla extract and the reserved sage, then gently mix until all apples are coated.Assemble the Pie
Preheat your oven to 400°F (200°C). On a floured surface, roll out one disc of dough to a 12-inch circle and transfer to a 9-inch pie dish. Pour in the apple filling, mounding slightly in the center. Roll out the second disc and either place it whole on top with slits cut for venting or create a lattice pattern.Apply Finishing Touches
Trim excess dough, then fold and crimp the edges. Brush the top with egg wash and sprinkle with coarse sugar if desired.Bake to Perfection
Place the pie on a baking sheet (to catch any potential overflow) and bake at 400°F for 20 minutes. Lower the temperature to 350°F (175°C) and continue baking for 35-45 minutes more, until the crust is golden brown and the filling is bubbling. If the edges brown too quickly, cover them with foil.Cool Completely
Allow the pie to cool on a wire rack for at least 3 hours before serving. This waiting period is crucial—it allows the filling to set properly and the flavors to fully develop.