1. Put the potatoes and celery root in a medium saucepan of salted water. Bring to a boil, reduce the heat, and cook at a low boil until tender, about 20 minutes.
2. Reserve 1/2 cup of the cooking water, and then drain the potatoes and celery root. Put them back into the saucepan along with 1/4 teaspoon salt and a pinch of pepper. Mash over very low heat, incorporating the reserved cooking water and 5 tablespoons of the butter.
3. In a large nonstick frying pan, heat 1/2 tablespoon of the oil over moderately high heat. Season the scallops with 1/4 teaspoon salt and 1/4 teaspoon pepper. Add half the scallops to the pan. Cook until browned, about 2 minutes. Turn and cook until browned on the second side and just done, about 1 minute. Remove. Heat the remaining 1/2 tablespoon oil in the pan and cook the remaining scallops. Remove.
4. Reduce the heat to moderate. Add the remaining 3 tablespoons butter to the pan and cook until golden, about 2 minutes. Add the scallops and any accumulated juices, the parsley, capers, lemon juice, and 1/8 teaspoon salt. Mound the puree on plates and top with the scallops and sauce.
Wine Recommendation: A white wine with full flavor makes a fine partner for scallops; on the other hand, a wine with some acidity is good, too. Try a crisp and minerally Chablis, made entirely from chardonnay, for the best of both.