To make the caramel:
Combine the sugar and water in a small heavy saucepan. It should have a thick, sludgy texture. Cover the pan.
Over medium heat, dissolve the sugar. You can swirl the mix as it heats, but DO NOT STIR. When the sugar has dissolved, increase the heat to medium-high.
When the syrup comes to a boil, remove the lid and attach a candy thermometer. Boil syrup until it reaches a deep amber color. Remove from heat.
Add the cream, taking care to avoid the sudden release of steam. Stir in the butter. Pour into a bowl and let cool.
To make the pie:
Mix together the apples, seasonings and thickeners.
Roll out your bottom crust and lay it in the pie pan. Spoon in about one-third of the apple filling. Reserve 1/4 cup to 1/2 cup of the caramel for drizzling on top of the pie and set that aside for later.
Drizzle a few tablespoons-full of caramel onto the apples. Spoon in another third of the fruit filling. Repeat the drizzle. Spoon in the rest of the fruit filling.
Weave your lattice strips to make the top crust. Roll the edges under and seal and crimp. Chill the pie in the freezer while the oven is heating to 425, a good 20 minutes.
Take the pie from the fridge and put it on a lipped, parchment-lined baking sheet.
Brush the lattice top lightly with egg white. Bake for 20 minutes.
Reduce heat to 375. Bake for another 45-55 minutes, until the crust is deep golden brown and sizzling, and you can see the filling bubbling.
Let the pie cool to room temperature.
Now it's time for the drizzle. I used a pastry bag and #4 tip for this; you can use a Ziplock bag with a teensy hole cut in one corner if you don't have pastry bags and tips. You can even try drizzling from a spoon, but that doesn't give you much control.
If your caramel has firmed up too much for easy spooning and drizzling, you might have to warm it in a glass bowl in the microwave. But take care not to heat it back up to a liquid state. Go easy on the heat; you want it barely warm, just soft enough to work with.
Drizzle the caramel horizontally, then vertically, across the pie, along and between the lattice strips. I piped some caramel into the openings between the lattice strips, too, so it kind of oozed into the filling.