Preheat oven to 375 degrees F. Grease a 9 by 13-inch baking dish with 1 tablespoon of the butter.
Brown the sausage in a large skillet over medium-high heat until it is cooked through. Transfer to a plate, while spooning off the excess grease, leaving about 2 tablespoons in the skillet. Add 4 tablespoons of the butter and melt. Add the onion, carrot and celery and saute until softened. Meanwhile, in a large mixing bowl, toss together bread cubes, saltines, poultry seasoning, parsley flakes, sage, salt and pepper. Stir the sauteed vegetables into the mixing bowl. Pour in the chicken stock and eggs and toss together. Turn the stuffing out into the prepared baking dish.
Bake in the preheated oven for 35 to 45 minutes until the top is golden brown and the juices in the stuffing are bubbling.
Preheat oil in the deep-fryer to 350 degrees F.
Cool the stuffing completely. Cut it into 12 squares. Remove each square from the dish and wrap it around a wooden stick, pressing it onto the stick with your hands. Make the stuffing form a log shape around the popsicle stick, leaving 2-inches of the stick exposed for the handle. This should look somewhat like a lumpy corn dog.
Roll the stick in the flour, coating it well and then shaking off the excess. Gently lower the sticks into the hot oil and fry until golden brown and crispy, about 5 minutes. Transfer to a paper towel lined plate to drain.