Place the bread cubes in a single layer on two quarter sheet trays (9-by-13-inch baking sheets) or countertop oven baking sheets and bake until dried out and crisp, about 15 minutes. Remove the bread cubes to a very large bowl.
Increase the oven temperature to 400 degrees F.
Meanwhile, heat the oil in a large saute pan over medium-high heat. Add the celery, onions and a pinch of salt and cook, stirring occasionally, until the vegetables are softened, 5 to 7 minutes. Add the sage, thyme, garlic and scallion whites and continue to cook until the vegetables and herbs meld together, 3 to 4 minutes. Transfer the vegetable mixture to the bowl with the bread cubes. Add the chestnuts and saltines and mix to combine well.
Whisk together the cream of mushroom soup and turkey stock in a medium bowl or large measuring cup. Pour over the bread cubes and vegetables. Season with salt and pepper and mix to combine well.
Transfer the bread mixture to a 9-by-9-by-2 1/2-inch broiler-safe baking dish. Using a wooden spoon, press down to compact the stuffing. Cover tightly with foil and bake for 30 minutes.
Remove the foil and switch the countertop oven to broil. Broil the stuffing until the top is golden brown and crispy, 2 minutes on high. Garnish with the scallion greens and Fried Sage, if using.
Heat the vegetable oil in a small saucepan over medium-high heat to 375 degrees F. To test if the oil is ready for frying, drop in a small sage leaf. If it begins to sizzle, the oil is ready. Add the sage leaves to the oil and fry until crisp, 20 to 25 seconds. Using a slotted spoon, remove the leaves to a paper towel-lined plate and sprinkle with salt.
You can fry the sage leaves in advance. They will keep in an airtight container at room temperature for up to 2 days.