Preheat oven to 350º. Spread bread cubes on two large baking sheets and toast for 30 minutes, stirring and rotating pans, until lightly browned and crisp. Transfer to very large bowl when done.
Meanwhile, in large skillet, melt 1 stick of butter. Pour half the melted butter in a bowl and reserve for later, to brush the top of the stuffing. To the pan with remaining melted butter, add celery, carrots and onion, a sprinkle of salt, and cook and stir over medium-high heat until softened and starting to brown, about 8 to 10 minutes. Transfer vegetables to bowl. Add sausage to skillet in chunks, and cook, breaking it up with wooden spoon, until cooked through and lightly browned, about 6 minutes.
Return vegetable mixture to the skillet and add the sage and thyme, a little salt and pepper, and cook for 1 minute. Add 1 cup of stock, cook and stir up any browned bits, until stock is reduced and cooked down by half, about 5 minutes.
Transfer sausage mixture to bowl with the toasted bread cubes. Add remaining 2 to 2½ cups of stock and toss well until bread cubes are evenly moistened. Taste and season with salt and pepper. Spread stuffing into large, buttered baking dish (about 3½ quart size) and brush with the reserved melted butter.
Bake in center of oven until heated through and nicely browned, about 1 hour, rotating casserole halfway through. Let stuffing rest 10 minutes before serving.