Mix eggs and milk together in a bowl until well beaten, then mix in salt and flour to make a stiff dough. On a floured surface roll dough out very thin. Let dry 30 minutes and then cut into thin strips. Place on greaseproof paper and let the noodles air dry.
In a large stockpot, add chicken stock, celery, onion, carrots and garlic. Simmer gently until vegetables are tender, about 1 hour.
Add chicken and optional sweetcorn. Season with salt, pepper and chicken stock cube. Simmer further 10 minutes, checking to see if liquid level has decreased. Add water if needed.
Bring soup to the boil, add noodles, stirring constantly to prevent sticking together. Return to the boil and cook the noodles until tender, about 20 minutes. Serve hot.