Heat oven to 400°F and butter a 2-quart baking dish.
Combine cheeses in a large bowl. Set aside one-third of the mixture. Add garlic, cream, and thyme. Season with salt and pepper. Add potatoes and toss gently to coat both sides.
Tightly arrange vertical potato slices in the baking dish, beginning with the perimeter and working toward the center, ensuring slices remain upright. Drizzle the remaining mixture over top, filling the baking dish halfway.
Cover with foil and seal the edges. Place onto a baking sheet to catch any overflow, then transfer to the oven. Bake for 30 minutes. Remove foil and bake for 30 minutes more, or until lightly browned.
Remove from oven and sprinkle remaining cheese on top. Return to the oven and bake for 30 minutes, or until the top is golden brown and crispy.
Remove from oven and cool for 10 minutes. Sprinkle with fresh chives before serving.