Preheat your oven to 400°F (200°C) and lightly grease a 9x9-inch baking dish. Place the peeled potatoes in a large pot of generously salted water and bring to a rolling boil over high heat.

Cook the potatoes until fork-tender, about 15 to 20 minutes. Drain well and return them to the warm pot, mashing them with the warmed milk, 3 tablespoons of butter, salt, and black pepper until completely smooth and creamy.

While the potatoes are boiling, heat the olive oil and remaining 1 tablespoon of butter in a large skillet over medium heat. Add the chopped onion and carrots, sautéing until softened and translucent, about 5 to 7 minutes.

Stir in the minced garlic and dried thyme, cooking until fragrant, about 1 minute. Add the shredded chicken, chicken broth, and heavy cream to the skillet, bringing the mixture to a gentle simmer.

Let the chicken mixture simmer for 3 to 5 minutes until the sauce thickens slightly and coats the chicken, then season generously with salt and black pepper. Transfer the chicken filling into the prepared baking dish, spreading it into an even layer.

Spoon the mashed potatoes over the chicken filling, spreading them gently to the edges to seal the casserole. Rough up the surface with the tines of a fork to create crispy peaks, and sprinkle the grated Gruyere cheese evenly over the top.

Bake uncovered for 20 minutes, or until the cheese is melted, bubbling, and beautifully golden brown on top. Let the casserole rest for 5 minutes before serving hot alongside a fresh green vinaigrette salad.