Preheat the oven to 350 degrees F. Spray a 9-by-13-inch baking dish with cooking spray. Line a baking sheet with parchment.
Tear the burgers (meat and buns) into quarters. Spread in a single layer on the prepared baking sheet. Bake until the buns are dry and starting to brown and the meat is heated through, 10 to 15 minutes. Let cool slightly, then transfer to a large bowl. (Leave the oven on.)
Heat a large skillet over medium-high heat. Add the butter and heat until melted, then stir in the celery and onion. Cook until the onion is softened and beginning to brown, 5 to 7 minutes.
Sprinkle the mustard powder, thyme and sage over the onion mixture and cook, stirring to combine, about 30 seconds. Pour in the broth a little at a time, using a wooden spoon to scrape up any browned bits from the bottom of the skillet.
Pour the broth mixture over the burgers in the bowl and toss until coated. Add the eggs and toss until combined. Transfer the mixture to the prepared baking dish and press into an even layer. Bake until the stuffing is cooked through and starting to brown on top, 40 to 45 minutes. Serve immediately or at room temperature.
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