Preheat the oven to 425 degrees.
Clean the outside of the squash well. Then, using a fork or sharp knife, poke 10 - 15 holes all around the squash. Bake for 45 - 60 minutes or until the squash is tender to the touch and you can stick a knife through with ease.
After the squash is cooked, let it cool for 10-15 minutes.
Cut the squash in half and remove the seeds and guts. Then use a fork to pull and scrape all of the spaghetti squash strands from the sides. You should get around 3 cups of spaghetti squash. Set the spaghetti squash aside.
In a 6 quart dutch oven, heat up the olive oil and butter over medium high heat. Once hot, add the garlic and cook until fragrant, about 2-3 minutes. Don't let the garlic burn.
Reduce heat to medium and then pour in the half and half and 1/4 of the chicken stock.
In a small bowl whisk together the remaining chicken stock and flour until completely dissolved and then whisk that into the dutch oven with the other ingredients.
Bring to a boil, and then reduce to a simmer and cook for 5-7 minutes until hot and bubbly and starting to thicken.
Stir in parmesan cheese and shredded chicken and cook until cheese is blended and chicken is hot, about 5 minutes. Taste for salt and add more if needed.
Stir in the spaghetti squash and cook until it's heated through, about 5 minutes.