Heat the oven to 350°F. Rub the bottom of a shallow 10x15-inch (or similar) rectangular baking dish with the butter. Sprinkle 1/4 tsp. salt over the bottom of the pan.
Cut the dark-green portion and all but about 1 inch of the light green off the top of the leeks. Peel away any tough or damaged outer leaves. Trim the ends by cutting the roots but leaving a bit of the base intact to hold the leek together. Cut each leek in half lengthwise. Gently wash each half under running water, fanning open the layers to rinse as thoroughly as possible. Pat the leeks dry and then arrange them cut side down in the baking dish. They should all fit snugly, but if they are crowded, turn a few on their sides. Sprinkle the thyme and 1/4 tsp. salt over the leeks.
Heat the cream and garlic in a small saucepan over high heat. As soon as the cream comes to a rolling boil (watch carefully and don’t let it boil over), remove the pan from the heat and let sit for 5 minutes. Pour the cream and garlic evenly over the leeks.
Cover the leeks with a piece of parchment cut to fit inside the pan.
Bake the leeks until the thickest ends are tender all the way through when pierced with a paring knife and the cream is almost entirely reduced, about 35 minutes. Sprinkle the leeks with the Parmigiano and salt to taste. Bake just until the cheese melts, an additional 1 to 2 minutes. Transfer the leeks to a warm serving platter.
Make Ahead Tips
You can wash and trim the leeks and arrange them in the buttered baking dish 6 hours ahead.
nutrition information (per serving):
Photo: Scott Phillips