Preheat oven to 350 degrees. Set spaghetti squash, cut side down, on a baking sheet. Roast for 30-40 minutes. Skin should pierce easily with a fork when done.
Remove from oven and allow to cool before flipping over. Using a fork, scrape the interior until spaghetti squash flesh easily comes out resembling long strands of pasta. Remove to a bowl and set side. Keep oven warm at 350 degrees.
In a medium skillet, melt butter and add spaghetti squash, tossing over medium-high heat. Continue until spaghetti squash has lost moisture and stops steaming, approximately 10 minutes. Toss in fine sea salt, freshly ground pepper and nutmeg. Remove from heat and set aside.
Cut tomatoes in half horizontally. Scoop out pulp. Fill tomato halves with spaghetti squash. Equally divide sharp cheddar cheese among tomatoes halves and place on a foil lined cookie sheet.
Bake until cheese has melted, approximately 10 minutes.
Remove and serve immediately.
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