Preheat oven to 400F. Drizzle olive oil over cut side of spaghetti squash halves and place upside-down on a baking sheet. Bake spaghetti squash at 400F 30-40 minutes or until tender. Remove from oven and once cool enough to handle, use a fork to shred inside of boats into spaghetti strands. Set aside.
In a medium saucepan over medium-high heat, saute garlic in butter until just fragrant. Whisk in flour until smooth, then gradually add milk and half-and-half, whisking constantly until combined. Bring sauce to a simmer over medium-high heat, stirring occasionally until thickened. Remove from heat and stir in nutmeg, Parmesan, chicken, spinach, and salt and pepper to taste.
Divide chicken alfredo sauce between the two spaghetti squash boats and gently toss with fork until filling is combined. Divide cheddar cheese between the two boats, sprinkling evenly on top.
Bake spaghetti squash boats an additional 15-20 minutes at 375F until cheese is melted and filling is bubbly. Enjoy spaghetti squash right away!