Preheat oven to 400F. Cut spaghetti squash in half lengthwise. Then, remove all the seeds and stringy parts. Place spaghetti squash cut-side-down on the baking sheet and use a fork to pierce holes all over the skin.
Flip the spaghetti squash over, spray the inside with cooking oil and season with salt and pepper. Flip the spaghetti squash cut-side-down again and transfer to the oven to bake for 25-35 minutes (time will vary based on size). The spaghetti squash is done when it easily strings apart with a fork and is al-dente when tasting.
Meanwhile, add cubed chicken to a large bowl and season heavily with salt. Let that sit at room temp while you prepare the remaining ingredients. This will help make the chicken more tender and flavorful.
Fill a medium-sized pot with water and bring it to a boil. Heavily salt the water. Once boiling, add the broccoli florets and let cook for about 3 minutes, or until vibrant green but still a little bit crunchy. Drain and set broccoli aside.
Bring a large pan to medium-high heat. Once hot, add enough olive oil to coat the bottom of the pan. Once the oil is hot, add the chicken. Spread out evenly in the pan but then don't touch it for about 2-3 minutes. Flip the chicken and cook for another 2-3 minutes on the other side or until nearly done through (it'll continue cooking in the oven later). Remove chicken from the pan and set aside.
Reduce heat to medium. Add a bit more olive oil. Then, add the onion. Sauté, stirring occasionally, for about 2-3 minutes. Then add garlic, stir, and cook for 1 minute longer.
Add in the butter and let it melt. Then, add in the flour and whisk constantly for 2-3 minutes or until the flour smell has gone away.
Slowly, add in milk little by little, stirring constantly. If it seems too thick, add in the additional 1/2 cup of broth (or milk), stirring constantly.
Reduce heat to a simmer and add in the cheese. Stir until it's become fully melted and smooth. Season with salt and pepper to taste.
***SEE NOTE - Once spaghetti squash is done roasting, use a fork to remove some of the strands. Add them to the cheese sauce along with the cooked chicken and broccoli. Toss to coat it in the sauce.
Stuff the mixture back inside of the spaghetti squash boats and top with a bit more shredded cheddar. Bake for another 10 minutes, then switch to broil for a few more minutes at the end to get it golden! Garnish with parsley & enjoy!