Preheat your oven to 375°F (190°C).
In a large skillet, melt 1 tablespoon of butter over medium heat. Add the onions and cook until they are translucent, about 5 minutes. Stir in the garlic and thyme, cooking for another 2 minutes until fragrant.
In a large mixing bowl, combine the sliced potatoes, carrots, and parsnips. Add the sautéed onion and garlic mixture, then season with salt and pepper. Toss everything to combine well.
In a separate bowl, whisk together the heavy cream and half of the Gruyère cheese until smooth. Pour this mixture over the vegetables and stir gently to combine.
Transfer the vegetable mixture into a greased baking dish, spreading it into an even layer. Top with the remaining Gruyère cheese and panko breadcrumbs drizzled with melted butter.
Cover the dish with aluminum foil and bake in the preheated oven for 30 minutes. Then, remove the foil and bake for an additional 20 minutes, or until golden and bubbly.
Once cooked, let the gratin cool for 10 minutes before serving. This resting time allows the layers to set and makes it easier to slice and serve.