Preheat. Turn on your favorite smoker and allow it to preheat to 275 degrees F using a mild hardwood like maple, alder, or pecan.
Season. Drizzle your cut and seeded acorn squash with olive oil. Use a brush or your hands to spread the oil across the entire cut surface of the squash. Season the inside of the squash with salt.
Smoke. Place the squash, cut side down, on a baking sheet lined with foil. Set the pan on the grates of your smoker, close the lid, and cook the squash for 1 hour.
Make the butter. In a medium-sized bowl, combine the butter, brown sugar, cinnamon, and nutmeg to make the brown sugar cinnamon butter.
Fill the squash. After the initial hour of smoking, carefully flip the squash over to expose the cut side. Fill the cavity of each squash with 1/4 of the brown sugar butter mixture. Use a basting brush to spread the butter across the exposed parts of the squash.
Baste the squash. Baste the squash with the butter mixture from the cavity every 30 minutes, cooking for an additional 2 hours, or until the squash reads between 200 and 210 F and pierces easily with a fork.
Serve. Remove the cooked squash to a serving platter with a large spoon. Allow your guests to scoop out squash onto their plates to get a good mix of the squash and butter, or cut the halves into quarters before serving.