Preheat oven to 350°F. Heat butter in a large sauté pan over medium-high heat until foamy. Add onions and celery, seasoning with 1 teaspoon kosher salt and ½ teaspoon black pepper. Cook, stirring occasionally, until tender but still slightly crisp, 8 to 10 minutes.
Add mushrooms and ½ teaspoon kosher salt and ¼ teaspoon black pepper. Cook until mushrooms are brown and liquid has evaporated, 12 to 15 minutes.
Add sage, paprika and nutmeg to the pan and cook, stirring constantly, until fragrant, 1 to 2 minutes.
Pour the mixture into a large bowl, then add diced burgers. Toss to combine. Add the chicken stock, tossing until the liquid has been absorbed. Pour the stuffing in a 9×13-inch baking dish.
Bake for 20 to 25 minutes, uncovered, until the top is brown and crispy.
