Grind the Earl Grey tea leaves to a fine powder using a spice grinder or mortar and pestle. Mix thoroughly with the granulated sugar and store in an airtight container for at least 4 hours (preferably overnight).
In a large bowl, whisk together flour, salt, and sugar for the crust. Add the cold, cubed butter and use a pastry cutter or your fingertips to work it in until the mixture resembles coarse crumbs with some pea-sized pieces remaining.
Drizzle in ice water, 1 tablespoon at a time, mixing gently until the dough holds together when pinched. Divide the dough into two equal portions, flatten into discs, wrap in plastic, and refrigerate for at least 1 hour.
In a large bowl, toss the sliced apples with lemon juice and vanilla extract. In a separate bowl, combine the Earl Grey infused sugar, flour, cornstarch, cinnamon, nutmeg, and salt. Sprinkle this mixture over the apples and toss until evenly coated. Let the mixture sit for 10-15 minutes.
Preheat your oven to 425°F (220°C) and place a baking sheet on the lower rack to catch any drips.
Roll one disc of dough on a floured surface into a 12-inch circle and carefully transfer to a 9-inch pie dish. Gently press into the bottom and sides, leaving a slight overhang.
Pour in the apple filling, mounding slightly in the center, and dot with butter pieces. Roll out the second disc and either place it whole on top (cutting vents) or create a lattice pattern. Seal and crimp the edges decoratively.
Brush the top crust with beaten egg and sprinkle with turbinado sugar. Cover the edges with foil or a pie shield to prevent over-browning.
Bake for 20 minutes at 425°F, then reduce the temperature to 375°F (190°C) and continue baking for 30-35 minutes until the crust is golden brown and the filling is bubbling.
Allow the pie to cool on a wire rack for at least 2 hours before serving to allow the filling to set properly.