1. Preheat oven to 450°F with rack in lower third of oven. Butter a 2-quart shallow flameproof gratin pan or baking dish (not glass.)
2. Cut fronds from fennel and chop enough to measure 1 tablespoon. Set aside. Cut off and discard stalks. Quarter fennel lengthwise and cut very thinly with slicer into a large bowl.
3. Peel potatoes and cut very thinly with slicer into bowl. Add butter with 3/4 teaspoon salt and 1/4 teaspoon pepper and toss to coat.
4. Transfer vegetables to gratin dish, spreading them out in an even layer, and cover tightly with foil.
5. Bake 30 minutes, then reduce oven to 350°F and continue to bake until vegetables are tender, 20 to 25 minutes more.
6. Remove from oven and remove foil, then sprinkle gratin evenly with cheese.
7. Preheat broiler and broil gratin 4 to 5 inches from heat just until cheese is lightly golden 4 to 5 minutes.
8. Let gratin stand 5 minutes before serving.