Preheat oven to 425°F. Roll out puff pastry so it's double the size of your Tarte Tatin dish. It should be rolled out enough to produce two circles the size of your pan. Cut the pastry in half.

Place the Tarte Tatin dish down onto one piece of the puff pastry to guide you, then remove the dish and cover the dented area with the grated cheddar cheese. Cover the cheese with the other piece of puff pastry, pressing down on the sides so you have a cheese pocket. Again, place the dish down onto the puff pastry, and using a knife, cut out a circle just slightly larger than it. Cut two large slits in the center of the pastry, then transfer to the refrigerator to keep cold while you assemble the rest of the tarte.

Heat olive oil in a small skillet over medium heat, then add sliced shallots. Season with Kosher salt and cook for 7-9 minutes, until shallots are softened and starting to brown. Remove from heat.

In the Tarte Tatin dish, warm ¼ cup white sugar with ¼ cup water and 1 teaspoon chili flakes over medium-high heat. When it starts to boil, begin lifting the pan and tilting it so you swirl the liquid. (Do not stir with a spoon or spatula—the sugar will stick.) Let it cook, swirling every 20-30 seconds, until it turns a color that resembles maple syrup. Remove from heat, add balsamic vinegar, then swirl again until it's evenly combined.