Preheat oven to 425°F.
In a medium saucepan, melt the butter on medium. Once melted whisk in the flour to form a roux.
Once combined, whisk in the water and the brown and granulated sugars.
Bring the mixture to a gentle boil, then reduce to a simmer for about 5 minutes. Remove from heat once the mixture is thick and combined.
While the mixture is cooling, peel, core, and thinly slice the apples. You can also roughly chop into chunks instead of slices if desired.
Place the apples in a large bowl and toss them with the cinnamon, cloves, allspice, and nutmeg. Pour the sugar mixture over the apples and stir to combine.
Place one pie crust in the bottom of the pie pan. Pour the apple mixture over the crust.
Place the top crust on in your desired pattern. If simply topping with crust, cut two air slits into the top crust.
Whisk together the egg and water. Brush the top of the pie crust with the egg wash and sprinkle with coarse sugar if desired. Place the pie in the oven to bake for 15 minutes.
After 15 minutes, lower the temperature to 350°F and bake an additional 45 minutes. If the top of your crust starts to get too dark, tent lightly with foil.
Let the pie cool for at least an hour until the filling is cooled and set.