Preheat your oven to 375°F (190°C).
Prepare the filling by sautéing celery and onions in a skillet over medium heat until softened.
In a large bowl, combine the sautéed vegetables, shredded turkey or chicken, thyme, garlic powder, and cheese.
Trim the crust off the loaf of French bread and cut it into slices that are roughly 1 inch thick.
Layer some of the filling onto each slice of bread and roll them up tightly.
Arrange the roll ups seam-side down in a greased baking dish.
Pour the broth over the roll ups, making sure they are well-saturated but not soggy.
Cover the dish with aluminum foil and bake for 25 minutes.
Meanwhile, make the creamy gravy by melting butter in a saucepan over medium heat.
Whisk in the flour, cooking for about 2 minutes until golden brown.
Slowly pour in the heavy cream and broth, whisking continuously until thickened.
Season the gravy with salt and pepper to taste.
Remove the foil from the roll ups and bake for an additional 10-15 minutes until golden brown.
Serve hot with a generous drizzle of creamy gravy over the roll ups.