Preheat oven to 350 degrees F.
Add one tablespoon of olive oil to a large skillet over medium-high heat. Once the pan is hot add the chicken. Season with salt and pepper. Cook the chicken for 5-6 minutes, stirring often. Remove the chicken to a clean plate and set aside.
Return the skillet to medium heat. Add the remaining tablespoon of oil and shallots to the skillet. Sauté the shallots for 2-3 minutes, or until soft and translucent. Add the garlic, salt, pepper, and Italian seasoning, and cook until fragrant- approximately 30 seconds.
Add the broccoli and 1-2 tablespoons of water to the skillet. Mix well and cover. Allow the broccoli to steam for 2 minutes before uncovering. Return the chicken back to the skillet. Continue to cook for 1-2 more minutes, stirring often. Remove from heat.
Meanwhile, in a large mixing bowl combine the cream cheese with the Greek yogurt. Add the chicken and broccoli to the cream cheese mixture, mixing well to combine. Stir in the parmesan cheese and half of the shredded mozzarella cheese with the chicken and broccoli.
Transfer the cooked strands of spaghetti squash to the mixing bowl with the chicken, broccoli, and cheeses. Gently mix to combine. Evenly divide spaghetti squash and chicken mixture between the spaghetti squash bowls. Top with mozzarella cheese (or cheese of choice) and place face up on the prepared baking sheet. Cover the baking dish and squash with foil, creating a tent to prevent the foil from touching the cheese.
Bake at 350 degrees for 15 minutes. Remove the foil tent and bake an additional 5-10 minutes, or until cheese is golden and bubbly.
Garnish with fresh chopped parsley or red pepper flakes, if desired.