In a large bowl, whisk together the flour, sugar, and salt. Add the cold, cubed butter and use a pastry cutter or your fingertips to work it into the flour until the mixture resembles coarse crumbs with some pea-sized pieces remaining.
Drizzle the chilled Irish cream over the mixture and stir gently with a fork. Add ice water, one tablespoon at a time, until the dough just begins to come together when squeezed (you might not need all the water).
Divide the dough into two portions (one slightly larger than the other). Shape each portion into a disk, wrap in plastic wrap, and refrigerate for at least 30 minutes or up to 2 days.
In a large bowl, toss the sliced apples with lemon juice. In a separate bowl, whisk together the sugars, cornstarch, cinnamon, nutmeg, and cloves. Add this mixture to the apples and toss until evenly coated.
Pour the Irish cream over the apples and gently mix. Let the mixture stand for 15 minutes, stirring occasionally.
Preheat your oven to 425°F (220°C). Place a baking sheet on the middle rack to preheat.
Roll out the larger disk of dough on a floured surface to a 12-inch circle. Carefully transfer it to a 9-inch pie dish, pressing gently against the bottom and sides. Trim the edges, leaving a ½-inch overhang.
Pour the apple filling into the crust, including any accumulated juices. Dot the top with small pieces of butter.
Roll out the remaining dough to a 10-inch circle. You can either place it whole over the filling or create a lattice top for a more decorative presentation.
Trim the top crust if needed, then fold the overhang of the bottom crust over the top crust edges. Crimp or flute the edges to seal.
Whisk together the egg and water, then brush this egg wash over the top crust. Sprinkle with turbinado sugar for a beautiful crystallized finish.
Cut a few small slits in the top crust if you're not using a lattice design. Place the pie on the preheated baking sheet.
Bake at 425°F for 20 minutes, then reduce the temperature to 375°F (190°C) and continue baking for 30-35 minutes more, until the crust is golden brown and the filling is bubbling.
Allow the pie to cool on a wire rack for at least 3 hours before slicing. This cooling period is crucial for the filling to set properly.