Place the potatoes in a large pot and cover them with cold water. Add a generous pinch of salt. Bring the water to a boil over high heat, then reduce to a simmer. Cook the potatoes until tender, usually 15-20 minutes. Drain the potatoes well. Removing as much water as possible is essential to avoid a watery purée.
While the potatoes are cooking, gently heat the milk in a small saucepan or the microwave until warm but not boiling.
Pass the cooked potatoes through a potato ricer or food mill back into the pot for the smoothest texture. If you don't have these, a regular potato masher will work, but the purée might not be as fine.
Over low heat, gradually add the butter to the potatoes, stirring continuously. The potatoes will start to become creamy and smooth. Slowly mix in the warm milk until the purée reaches your desired consistency. It should be silky and smooth—season with salt, pepper, and nutmeg to taste.
The most dramatic part of this dish is the presentation. Bring the cocotte directly to the table. Make sure your guests are present as you break the pastry seal. This releases the fragrant aromas of garlic and thyme, creating an unforgettable sensory experience.
While cutting the chicken into serving pieces, reduce the cooking juices by 50%. Serve the chicken drenched in the sauce with the garlic still in its jacket. Pass a green salad and potato puree separately.