Preheat oven to broil.
Spray a large non-stick skillet with oil and heat over medium high heat.
When the skillet is hot add in the chicken, season with salt and pepper and sauté until cooked through.
Remove the chicken from the skillet onto a plate.
Turn the heat down to medium and add a couple teaspoons of olive oil to the skillet.
Swirl the oil to coat the bottom of the skillet then add in the shallot, garlic and red pepper flakes.
Sauté for about a minute then add in the broccoli florets.
Season with salt and pepper then add in a couple tablespoons of water and cover with a lid.
Let the broccoli steam for about 2 minutes, then uncover and add the chicken back in.
Add in the cream cheese and chicken broth and stir everything until smooth and a sauce starts to form.
Add in half of the shredded cheese and stir until melted.
Remove from the heat and stir in the Greek yogurt until combined.
Taste for seasoning then pour the filling into the bowl with the spaghetti squash strands and stir together until combined.
Divide the filling evenly into the spaghetti squash shells and top with the remaining shredded cheese.
Broil in the oven until the cheese is melted then serve.