Preheat oven to 400 degrees. Rub each half of the squash, inside and out, with a teaspoon of olive oil and season with salt and pepper. Place upside-down on a foil-lined baking sheet, with one sprig of thyme tucked into the cavity of each half. Roast for about 45 minutes, until a fork poked into the non-hollow end of the squash slides in easily. Remove and let cool just slightly.
Scoop out the flesh of the squash halves and add to the bowl of a food processor (you can also mash by hand with a potato masher or fork, it will just be a bit chunkier). Add 3/4 cup of the cheese, the leaves from the remaining thyme sprig, nutmeg, and salt. Pulse until blended, then pour in the milk or half and half while the machine is running.
Transfer mixture to a small gratin dish (about 8 inches long) or a shallow pie plate. Sprinkle on the remaining 1/4 cup cheese and return to the oven for about 20 minutes. After 15 minutes, turn on the broiler so that the top of the dip gets brown. Serve with crackers, toast, or pita chips.