For Fritters: In a small saucepan, combine 2 tablespoons plus 2 teaspoons water, milk, butter and pinch salt. Bring to a boil over medium heat, then add flour and, using a wooden spoon, stir until batter pulls away from sides of pot and forms a ball. Spread the batter on a plate and let cool, then transfer to a bowl and, using a fork, blend in cheese and egg. Cover surface of dough with a piece of plastic wrap and chill in the refrigerator for 30 minutes.

For Soup: While dough is chilling, cut tips from asparagus; thinly slice and set aside. Thinly slice stems. Heat butter in a wide heavy 4- to 5-quart pot over medium heat until melted. Add asparagus stems and shallot; cook, stirring occasionally, until asparagus is tender, about 10 minutes. Stir in potatoes and cook for 1 minute, then add broth. Cook at a gentle simmer, partially covered, until potatoes are tender, about 25 minutes. In batches, carefully purée soup until smooth; return to pot and season with salt and pepper.

Using floured hands, divide dough into 2 equal pieces. On a lightly floured surface, roll 1 dough piece into a 10-inch log. Cut log crosswise into 20 (1 ⁄ 2-inch) pieces, then roll pieces into balls. Repeat with remaining dough.

Fill a medium pot with about 1 inch oil. Heat oil to 365°. In batches of 10, fry fritters until golden, about 2 minutes. Using a slotted spoon, transfer to paper towels to drain.

Gently warm soup, then divide among bowls. Top with fritters and sliced asparagus tips. Sprinkle with parsley and drizzle with oil.