Preheat oven to 375 degrees F. Butter a 2 quart baking dish. Cut off the entire rough exterior of the celery roots, removing all the brown parts. Cut each celery root in half lengthwise then cut into ¼ inch wide pieces. Set aside. Cut off the leaf and dark green parts of each fennel bulb, leaving just the bulb. Cut each bulb in half lengthwise and use a mandolin to slice 1/4 inch thick slices. In a vegetable steamer, combine the celery root and fennel, and steam for 5 minutes. Remove from the steamer and set aside to cool slightly. Then using your mandolin, slice the apples into ¼ inch thick rings.
To assemble gratin, place 1/3 of the celery root on the bottom of the prepared dish. Sprinkle celery root with red pepper flakes, salt and pepper. Overlap with 1/3 of apple slices and sprinkle with salt. Put 1/3 of fennel over the apples and once again sprinkle with red pepper flakes, salt and pepper. Repeat process two more times until all celery, fennel, and apple are used. Pour cream around the sides of the dish until the cream reaches halfway up the dish. Sprinkle the top of the gratin with cheese. Cover and bake for 35 minutes, or until the vegetables and apple feel tender. Turn the oven to broil, remove the cover, and broil until the gratin starts bubbling and turning golden brown, about 3 minutes. Let cool 15 minutes to allow the cream to firm up and serve.