Position a rack in the center of the oven and preheat to 400°F. Lightly oil an 11 1/2 x 8-inch (2-quart) baking dish. Bring a large pot of lightly salted water to a boil over high heat.
Process all the pesto ingredients in a food processor fitted with the metal blade until the mixture forms a coarse paste. (The pesto can be stored in an airtight container and refrigerated for up to 1 month).
Peel the potatoes and cut into 1/8-inch-thick rounds (a mandoline or food processor does the quickest work). Add to the boiling water and cook until barely tender, about 8 minutes. Drain and toss with the pesto to coat evenly. Season to taste with salt and pepper.
Layer half of the potatoes in the dish, sprinkle with one third of the cheddar cheese, and sprinkle with the 2 Tbsp. basil or 1 Tbsp. marjoram. Continue until you have 3 layers, finishing with cheese. Pour in the hot broth. Cover tightly with aluminum foil.
Bake for 30 minutes. Uncover and continue to bake until the cheese is bubbling and starting to brown, about 15 minutes. Serve hot, sprinkled with parsley.