Position a rack in the middle of the oven and preheat the broiler. Brown the corn muffin pieces on a baking sheet, tossing once, for about 5 minutes.
In a large skillet, heat the EVOO, 1 turn of the pan, over medium-high heat; melt in the butter. Add the apple, onion and celery; season with salt and pepper. Cook until crisp-tender, about 5 minutes. Add the muffin pieces, parsley, poultry seasoning and thyme. Moisten with the chicken stock to your liking, 1 to 1 1/2 cups. Turn off the heat.
Mix the cranberry sauce with the vegan (or regular) mayo and slather it on the roll tops. Assemble the sandwiches on the roll bottoms with the cornbread stuffing, sliced turkey and cheese. Broil to melt the cheese. Top with the onion, cornichons and roll tops.