In a medium bowl, whisk the eggs with the milk, chives and parsley. Add the flour, cornmeal, cayenne, sage and 1 teaspoon of salt; stir until evenly moistened.
Heat 1 tablespoon of the oil in a large stockpot. Add the leeks and carrots and cook over moderate heat, stirring occasionally, until crisp-tender, about 8 minutes. Add the stock and bring to a simmer. Cook over moderate heat until the carrots are tender, about 5 minutes.
Meanwhile, heat the remaining 1 tablespoon of oil in a medium skillet. Add the mushrooms and cook over high heat until browned, about 5 minutes. Add the mushrooms and the peas to the soup. Season with salt and pepper. Let simmer over moderate heat.
Spread 1/2 cup of the noodle batter onto the center of an inverted cake pan. Using a large offset spatula, scrape 1/4-inch strips into the soup, creating long, thin noodles. Repeat with the remaining batter. Gently stir the soup to separate the noodles and cook until tender, about 2 minutes. Add the turkey and cook just until hot, then serve.