Preheat oven to 400°F.
Use a large knife to cut the spaghetti squash lengthwise from stem to tail. Lightly coat the flesh with cooking spray, and place cut-side down on a baking sheet. Bake for 30 to 40 minutes, under tender. Let sit until cool enough to handle.
Meanwhile, heat oil in a large skillet over medium heat. Add shallots and garlic; cook 3 minutes, until soft. Whisk in flour, and cook 2 minutes. Gradually stream in milk, and whisk until smooth. Gently simmer until thickened, about 3 to 4 minutes. Whisk in lemon juice, Parmesan cheese, salt, and pepper. Remove from heat and stir in shredded chicken.
Scrape out and discard the seeds of the spaghetti squash. Use a fork to scrape out all of the spaghetti-like strands, and stir them into the chicken Alfredo mixture.
Fill empty squash boats evenly with chicken Alfredo mixture, and sprinkle mozzarella cheese overtop. Return to oven and bake for 10 minutes. Set broil to HIGH and broil for 2 minutes, or until cheese is golden in some spots. Remove from oven and garnish with fresh parsley.