Melt butter and in a large sauce pan. Add onions, celery and carrots and saute over medium heat until onion are translucent, about 5 minutes.
Whisk 3 Tablespoons corn starch into one cup of chicken broth, then add to remaining 2 cups broth. Add broth to sauteed vegetables and bring to a simmer. Cook over medium low heat until slightly thickened, about 5 minutes.
Add poultry seasoning, thyme, peas, mushrooms, and chicken or turkey to veggies and simmer on low until carrots are tender. Add salt and pepper to taste.
Fill individual ramekins and top with mashed potatoes and grated cheese.
Bake at 350 degrees F. until filling is hot and bubbly and cheese is melted, about 15-20 minutes.