Preheat the pellet smoker, covered grill, or oven to 425°F. If using a charcoal or gas grill, establish an indirect cooking zone.

Spread the frozen tater tots in a single layer on a large rimmed sheet pan. Cook for 30 to 35 minutes, rotating the pan halfway through, until the tots are deeply golden and firmer than you would normally serve them. Remove the pan and reduce the cooker to 375°F. Reserve about one-third of the best-looking tots for the top.

While the tots cook, place a large skillet over medium heat. Add the sage sausage and cook for 7 to 9 minutes, breaking it into bite-size crumbles, until browned and cooked through. Transfer the sausage to a plate, leaving about 1 tablespoon of rendered fat in the skillet.

Add 4 tablespoons of the butter to the skillet. Stir in the onion, celery, apple, salt, and black pepper. Cook for 7 to 9 minutes, stirring occasionally, until the vegetables soften and the apple is just tender.

Add the garlic, sage, and thyme. Cook for 1 minute, stirring, until fragrant. Remove the skillet from the heat.

Use the remaining 2 tablespoons butter to grease a 12-inch cast-iron skillet or a 9-by-13-inch baking dish.

In a large bowl, whisk the eggs, chicken stock, and Dijon mustard until smooth. Add the cooked sausage, vegetable mixture, dried cranberries, and about two-thirds of the crisp tots. Fold gently with a wide spatula just until evenly coated.

Transfer the mixture to the prepared skillet or baking dish. Arrange the reserved tater tots over the top. Cover tightly with aluminum foil.

Cook at 375°F for 20 minutes. Remove the foil and continue cooking for 15 to 20 minutes, until the top is crisp and the center reaches 160°F on an instant-read thermometer.

Rest for 10 minutes. Garnish with fresh sage or thyme, if desired, and serve warm.