Preheat the oven to 425°. On a piece of lightly floured parchment paper, gently roll out the puff pastry, smoothing out any wrinkles, until ¼-inch thick. Cut the pastry into a 10-inch diameter circle. Place the parchment onto a baking sheet and place in the fridge to chill.
In a deep 10-inch cast-iron skillet over medium heat, melt the butter. Stir in the sugar and cook until an amber color develops, 6 to 8 minutes. Add the apples and cook, tossing occasionally, until you have a dark caramel sauce and the apples have shrunk slightly, 20 to 25 minutes. Flip the apples so the cut side is facing up and remove from the heat.
Sprinkle 1 heaping tablespoon of Tillamook Medium Cheddar Cheese onto the center of each apple. Sprinkle the salt and pepper over the top and place the pastry over the cheese, tucking it over the apples. Bake until the crust is puffed and golden brown, 8 to 10 minutes. Reduce the heat to 350° and bake until the apples are tender and cooked all the way through, 30 to 35 minutes longer. Remove from the heat and let cool for 5 to 10 minutes. Gently run an offset metal spatula around the crust to release any caramelized edges. Place a large plate over the skillet and carefully invert the tarte onto the plate. Serve warm.