Preheat oven to 375 degrees F. Butter a 2-quart gratin dish or other ovenproof casserole.
Cut off the entire rough exterior of the celery roots, making sure that all brown parts are removed. You will be removing about 1/4 to 1/2-inch of the exterior of each. Next, cut each celery root in 1/2 lengthwise. Turn one 1/2 cut side down, and cut into 1/4-inch wide pieces. Repeat with the remaining celery root. Set aside.
Cut off the fronds and the dark green parts of each fennel bulb, leaving just the bulb. Cut each bulb in half lengthwise. Cut each half lengthwise into 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.
Cut the apples into 1/4-inch-thick rings.
Place 1/3 of the celery root on the bottom of the prepared dish. Sprinkle with salt and pepper to taste. Overlap with 1/3 of the apple slices, and sprinkle with a little more salt but no pepper. Put 1/3 of the fennel over the apples, and once again sprinkle with salt and pepper. Repeat with the remaining celery root, apple, and fennel. Pour cream around the sides of the dish until the cream reaches about halfway up the dish. Sprinkle the top of the gratin with the cheese. Cover and bake for 35 minutes, or until the vegetables and apple feel tender when pierced with a fork or a skewer. Turn the oven to broil, remove the cover, and broil until the gratin is golden brown and bubbling, about 3 minutes.
Let cool 15 minutes to allow the cream to firm up slightly and serve.