Cut slivers off top and bottom of spaghetti squash and discard. Cut remaining squash into 1" thick rings. Run a knife around the center of each ring to remove seeds.
Line a baking sheet with parchment paper and drizzle paper with olive oil. Swirl the squash rings around in the olive oil, flip the rings over and repeat until they are lightly coated on both sides. Sprinkle kosher salt and pepper on both sides.
Roast in the oven for 30 minutes and remove to cool for about 10 minutes.
While squash is roasting, soak dried mushrooms in warm water to reconstitute for 30 minutes. Drain, rinse, and pat dry. Roughly chop.
In a large sauté pan, melt butter over medium heat. Add mushrooms and sauté for about 5 minutes. Add walnuts and cook for another minute or so to toast.
Add kale, cream or half-and-half, thyme and oregano, and nutmeg and cook until kale is wilted and cream is reduced.
After squash is cool enough to handle, peel off the skin on each ring and discard. Pull apart the spaghetti strands with your hands, and toss with the sauce.
Divide onto plates and top with a few crumbles of goat cheese.