French Fries with Gravy and Cheese

Peel potatoes, then cut lengthwise into 1/4-inch-thick sticks and submerge in a bowl of ice and cold water. Rinse potatoes in several changes of cold water until water is clear. Drain in a colander, then spread potatoes in 1 layer on several layers of paper towels and pat very dry. Heat about 8 cups vegetable oil in a 5- to 6-quart heavy pot over moderate heat until a deep-fat thermometer registers 375 °F. Put oven rack in middle position and preheat oven to 200 °F. Cook shallot with thyme in butter in a 1- to 1 1/2-quart heavy saucepan over moderate heat, stirring frequently, until shallot is softened and golden, about 2 minutes. Add wine and boil until reduced by half, about 3 minutes, then discard thyme sprigs. While wine reduces, stir together water and cornstarch until cornstarch is dissolved. Stir 1 cup beef or veal demi-glace into wine and bring to a boil. Whisk in cornstarch mixture and return to a boil, then boil until sauce is slightly thickened, 2 to 3 minutes. Remove from heat and stir in black pepper, salt, and unsalted butter until butter is melted. Once oil is ready, increase heat to moderate and fry potatoes in 4 batches, stirring occasionally, until deep golden, 5 to 6 minutes per batch, returning oil to 375 °F between batches. Transfer fries with a slotted spoon to a baking sheet lined with several layers of dry paper towels and sprinkle lightly with salt. Keep fries warm in oven while frying remaining batches. Put 4 ovenproof plates in oven, divide fries among plates, and sprinkle with cheese. Heat until cheese is just warmed through, about 2 minutes. Stir chives into sauce and drizzle over fries. Serve immediately.