Boil the cubed potatoes in a large pot of salted water until fork-tender, about 15 minutes, then drain well.

Mash the potatoes with 3 tablespoons of butter, warm milk, minced garlic, salt, and pepper until creamy and smooth.

Preheat your oven to 400 degrees F (200 degrees C) and lightly grease a 9x13-inch baking dish.

Heat the olive oil and remaining tablespoon of butter in a large skillet over medium heat, then saute the sliced leeks and diced carrots until softened, about 6 to 8 minutes.

Sprinkle flour over the vegetables, stir well to coat, and cook for 1 minute before gradually whisking in the chicken broth and heavy cream.

Bring the mixture to a simmer, then stir in the shredded chicken and fresh thyme, cooking until the sauce thickens into a rich gravy, about 5 minutes.

Spread the chicken and vegetable filling into the prepared baking dish in an even layer.

Carefully spread the mashed potatoes over the chicken filling, fluff the top with a fork to create ridges, and sprinkle evenly with the Gruyere cheese.

Bake for 20 to 25 minutes until the cheese is melted, the top is golden brown, and the edges are bubbling. Let rest for 5 minutes before serving with a crisp green salad.