Butter a 13x9x2-inch baking dish (3-quart) and set aside.
Spread torn bread pieces in an even layer on a rimmed baking sheet; toast in oven, tossing occasionally, until very dry and beginning to brown, 20 to 25 minutes. Transfer to large bowl and let cool. Increase oven temperature to 375 °F (191 °C).
Meanwhile, heat olive oil in a large skillet over medium heat. Add sausage; cook and stir, breaking up with a wooden spoon, until browned and just cooked through, 7 to 8 minutes. With a slotted spoon, transfer sausage to a clean bowl, leaving the fat in the pan. Add mushrooms, salt and pepper to pan; cook and stir until mushrooms are tender and their juices have mostly evaporated, 7 to 8 minutes. Add leeks and celery; cook and stir until softened, about 5 minutes. Stir in sage and thyme; check for seasoning and add more salt and pepper to taste. Remove from heat.
Add sausage with accumulated juices and mushroom mixture to toasted sourdough in bowl; drizzle chicken broth over and gently toss to combine. (recipe note #2)
Transfer bread mixture to prepared baking dish and spread into an even layer; drizzle evenly with melted butter. (recipe note #2) Bake uncovered until golden brown, 35 to 40 minutes. Serve.
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Notes
If you prefer very moist stuffing, add 1 to 1 1/2 cups more broth, but not so much that the mixture looks soupy.
At this point you can tightly cover and freeze if desired for up to 1 month and bake from frozen, making sure it reaches an internal temperature of 165˚F before serving. You can also refrigerate after cooking for 3 to 4 days and reheat or freeze for up to 1 month. To reheat refrigerated stuffing, remove from refrigerator and let stand 30 minutes at room temperature. Reheat in a 350˚F oven, covered with foil for the first 25 to 30 minutes, and then uncovered until hot all the way through with crispy edges, 10 to 15 minutes more.