In a saucepan, bring 1 1/2 quarts of salted water to a boil. Add fiddleheads and return to a boil. Using a slotted spoon, transfer fiddleheads to an ice bath and chill. Drain and pat dry, removing as much of the outer brown, tissue-like membrane as possible.

In a skillet saute shallots in butter until softened, about 2 minutes. Add thyme, morels, and garlic and continue to cook until morels have softened and given up their liquid, about 3 minutes. Continue to cook until almost all liquid is evaporated, about 2 more minutes. Add chicken stock and cook until reduced by half. Add fiddleheads and cook 2 minutes, add cream, chives, and parsley, and season with salt and freshly ground black pepper, to taste. Serve immediately, garnished with Parmesan curls.