1. With a fork, stir chicken to mix well with 1 1/2 teaspoons cornstarch, egg, 1/4 cup broth, salt, grated lemon peel, and 1 tablespoon fresh dill (or 1 teaspoon dried).
2. Bring 2 cups broth to simmering in a 10- to 12-inch frying pan. As broth heats, drop level tablespoon portions of chicken into the liquid, keeping pieces slightly apart. Cover and simmer until meat is white in center (cut to test), 3 to 4 minutes.
3. With a slotted spoon, transfer chicken to a bowl. Mix remaining 1 tablespoon cornstarch with 2 tablespoons water until smooth, and whisk into broth. Turn heat to high, add cheese, and whisk until it melts.
4. Return chicken to sauce, mix gently, and heat for about 1 minute. Pour over noodles in a bowl and sprinkle with remaining dill.