Place the flour, sugar and salt in a food processor. Close lid and pulse a few times. With the processor on, gradually add the cubed butter. Then add the ice water. As soon as a dough ball forms, turn off the machine. (If you don't have a food processor, you can cut the butter into the flour mixture in a bowl with a pastry blender or a fork. Then stir in the water until completely combined. Knead with hands to form the dough.)
Roll out half of the dough on a lightly floured surface to create a 12-inch disk, about ⅛" thick. Gently place in pie plate. Next make the filling.
For the filling
Preheat oven to 450°F.
Place sliced apples in a large bowl. Sprinkle with lemon juice and toss together.
In another bowl, stir together remaining filling ingredients (except butter). Pour over apples and stir to combine. Pour filling into prepared crust. Dot with 2 tablespoons butter.
For the lattice crust
Roll out remaining dough as directed above. Use a pizza cutter to cut dough into 8 strips. Create a lattice pattern on top of pie. If desired, use one of the strips to cut into 3 pieces and make a braid. Use the braid as one of the strips to weave into the lattice. Fold the edges under then use your fingers to crimp edges or you can use a fork to press around the edges if that's easier. Brush with egg and sprinkle with sugar.
Place pie on a cookie sheet with raised edges (I used a half sheet pan) and put in preheated 450°F oven for 10 minutes. Turn oven down to 350°F and continue baking for 40 minutes. Let cool slightly and serve with ice cream, if desired. Store covered in the refrigerator for up to 5 days.
Note: If you have extra pie dough, roll out and cut into circles with a biscuit cutter. Dip in cinnamon sugar and bake at 350°F. for 10 minutes.