Using a large, sharp knife, pierce the spaghetti squash in several places.
Place the spaghetti squash in a glass baking dish and cook in the microwave on HIGH for about 12 minutes, turning the squash halfway through cooking.
Before handling, let the squash stand for 10 minutes. Cut in half lengthwise and scrape out the seeds and fibers. Using a fork, twist out strands of the spaghetti squash flesh and place in a large bowl.
While the spaghetti squash is cooling, heat a large nonstick skillet over medium high heat. Lightly coat with cooking spray and add the chicken. Cook, stirring occasionally, until the chicken is cooked through. Transfer the chicken to a bowl. Set aside.
Lower the heat to medium, and add the olive oil and butter to the skillet.
Add the shallots and cook until tender, about 3 minutes. Stir in the garlic, rosemary, thyme and oregano. Cook, stirring, for 1 minute.
Stir in the chicken broth. Allow the mixture to simmer for 2 to 3 minutes to allow the flavors to meld.
Add the spaghetti squash and cooked chicken to the skillet and toss with the sauce. Add the grated Parmesan cheese and parsley, and toss again. Taste and season with salt and pepper, if desired. Serve.