Melt butter at medium heat in large dutch oven. Cook onions, carrots, and celery, until onions are translucent.
Stir in flour to coat vegetables, then slowly stir in chicken stock and milk. Add turkey, green beans, and spices; salt to taste.
Heat to a boil, then reduce to a simmer, let sauce thicken.
In a large bowl, whisk together flour and salt. Add mashed potato and egg, using a fork to combine. Knead gently in the bowl until a dough comes together. It will be sticky, but if wet to the point of falling apart (which may happen if you added milk or cream to your mashed potatoes), add more flour, a tablespoon at a time, until a soft, scoopable dough forms.
Drop by tablespoonfuls onto simmering stew. Cover and simmer for 20 minutes.