Peel, halve and core the apples. Cut them into 3/4 inch wedges. Toss the apples in a large bowl with the juice of the lemon.
Melt the butter in a 12" skillet. Stir in the brown sugar. Add the apples and turn to coat. Increase the heat to high and cook, turning occasionally, until most of the wedges are tender but not mushy, about 15 minutes. Do not overcook.
Immediately scoop the apples and their juices onto a rimmed baking sheet. Sprinkle the flour and cinnamon over them and toss until the flour disappears. Place the baking sheet on a wire rack and let the apples cool to room temperature, about 40 minutes.
Preheat the oven to 450. Arrange the apples in the pie shell; drizzle the juices over them. Cover with top crust and crimp. Dissolve the sugar into the milk and brush the top crust with a pastry brush. Prick with a fork.
Bake the pie for 25 minutes, or until the crust is golden brown. Reduce the heat to 375 and bake for 20 minutes longer, or until you hear the filling bubbling.
Transfer the apple pie to a wire rack and let cool for at least 1 hour before serving.