Spread bread cubes in a single layer on a large rimmed cookie sheet and bread crumbs on a small rimmed cookie sheet; let dry several hours or overnight. Adjust oven rack to lower-middle position and heat oven to 400 degrees. Bake bread cubes until toasty, 12 to 15 minutes. (Do not toast crumbs.) Reduce oven temperature to 350 degrees. 

Meanwhile, heat a splash of oil over medium-high heat in a 12-inch skillet; add sausages and cook, turning once, until crisp and brown, 5 to 7 minutes. Remove sausages from skillet, cool slightly, and then cut them into small dice. Return skillet to medium-high heat and add the oil. Add onions and celery; sauté until soft, 8 to 10 minutes. Mix bread, sausages, sautéed onions and celery, raisins, parsley, sage, thyme, salt and pepper in a large bowl. Whisk broth and eggs (or egg replacer); add to stuffing mixture and toss to coat.

Turn into a 3-quart greased baking dish (a 13- by 9-inch works well). Cover with foil and bake until steamy hot, about 30 minutes. Remove foil; continue to bake until top is crusty, about 10 minutes longer. Serve.