In a large stock pot over medium heat, melt 4 tablespoons butter. Add the shallots, season with salt, and sauté, stirring occasionally, until soft and translucent, 6–8 minutes. Remove fennel stalk and fronds from bulb; save for another use or discard. Cut bulb in half, then cut out and discard the core from each half. Slice fennel into a ½-inch dice and set aside.Using a knife to peel off the tough skin of the celery root; discard the skin. Slice celery root into a ½-inch dice. To the pot, add fennel, celery root, apple, and walnuts; stir to combine. Sauté over medium heat, stirring occasionally, until tender, 3–4 minutes. Season with salt and pepper.
Turn off the heat; add the bread and raisins to the pot. Stack sage leaves, then roll like a cigar and thinly slice across to create fine ribbons of sage (also known as a "chiffonade"). Remove thyme leaves from the stem by pinching the tip of the sprig and pulling straight down. Add coriander, sage, and thyme leaves to the pot; season generously with more salt and pepper.
Add milk a little at a time, and use your hands to gently combine all the ingredients after each addition. (If mixture is too hot, gently mix with wooden spatula or spoon until the stuffing cools.) Make sure each crouton is coated with milk, but not soggy. When enough of the milk has been added, allow the stuffing to rest, stirring occasionally, until the milk has fully absorbed, about 15 minutes. Meanwhile, grease a 13-by-9-inch baking dish with 1 tablespoon butter and preheat oven to 350 F.
Transfer the dressing to the prepared baking dish and bake until browned and crispy, about 50-60 minutes.