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Adjust oven racks to lower-middle and upper-middle positions. Preheat oven to 275°F. Spread bread evenly over 2 rimmed baking sheets. Stagger sheets on oven racks and bake until bread is completely dried, about 50 minutes total, rotating sheets and stirring bread cubes several times during baking. Remove from oven and allow to cool. Increase oven temperature to 350°F.
In a large Dutch oven, melt butter over medium-high heat until foaming subsides, without allowing butter to brown, about 2 minutes. Add sausage and mash with stiff whisk or potato masher to break up into fine pieces (largest pieces should be no bigger than 1/4 inch). Cook, stirring frequently, until only a few bits of pink remain, about 8 minutes. Add onion, celery, fennel, garlic, and thyme and cook, stirring frequently, until vegetables are softened, about 10 minutes. Remove from heat and add half of chicken stock.
Whisk remaining chicken stock, eggs, tarragon, and 3 tablespoons parsley in a medium bowl until homogeneous. Stirring constantly with a wooden spoon, slowly pour egg mixture into sausage mixture. Add bread cubes, oysters, and oyster liquor and fold gently until evenly mixed. Season lightly with salt and pepper.
Transfer stuffing to a buttered 9- by 13-inch rectangular baking dish (or 10- by 14-inch oval dish) and bake until browned on top and an instant read-thermometer reads 150°F when inserted into center of dish, about 45 minutes. Remove from oven, let cool for 5 minutes, sprinkle with remaining parsley, and serve.