Preheat oven to 325F with racks in the upper middle and lower middle positions
In a bowl, toss the celery with 1 tbsp of the butter and set aside
In a food processor, combine shallots, sage, thyme, salt, pepper, and remaining butter. Process to form a smooth paste, about 30 seconds, scraping the bowl as needed
In a large bowl, combine the bread and shallot-herb paste, tossing gently
Fold in the celery then divide the mixture between 2 rimmed baking sheets. Bake until the celery is tender and the bread is crisp and golden, 50 - 60 minutes, stirring the bread and switching and rotating the pans halfway through. Let cool slightly
At this stage, the bread mixture can be cooled, bagged, and stored for a day
When ready to proceed, increase the oven temperature to 400F
Transfer the bread mixture to a large bowl, scraping any browned bits off the sheet pans. Fold in the broth, cream, and parsley. Let sit for 10 minutes, stirring occasionally
Transfer to a 13 x 9 inch baking dish and spread evenly. Bake on the upper middle rack until well browned on top, 40-45 minutes, rotating the dish halfway through. Let sit 20 minutes before serving.