Step 1

Adjust oven racks to lower-middle and upper-middle positions. Preheat oven to 425°F. Spread cornbread evenly over 2 rimmed baking sheets. Stagger sheets on oven racks and bake until lightly toasted, about 10 minutes. Remove from oven and allow to cool.

Step 2

In a large Dutch oven, melt butter over medium-high heat until foaming subsides, about 2 minutes, without allowing butter to brown. 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.

Step 3

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 cornbread cubes, oysters, and oyster liquor and fold gently until evenly mixed. Season lightly with salt and pepper.

Step 4

Transfer dressing to a buttered 9- by 13-inch rectangular baking dish or 10- by 14-inch oval dish. The dressing can be covered with aluminum foil and refrigerated for up to 2 days at this point (the flavor will improve as it sits). When ready to bake, preheat oven to 375°F. Uncover dressing and bake until an instant-read thermometer reads 150°F when inserted into center of dish and dressing is crisped on top, about 45 minutes. Remove from oven, let cool for 5 minutes, sprinkle with remaining parsley, and serve.