Preheat oven to 375°F and grease a large casserole dish or a 9 x 13’'” baking dish.
After the spaghetti squash has been roasted, scrape out the strands and transfer it to a large bowl lined with a kitchen towel or a few layers of paper towels. This step will help absorb some of the extra liquid.
Remove the towel from the bowl and then add the yogurt, cheese (reserve a little for topping), egg and seasonings. Stir together until well-combined and set aside.
Once the oven is preheated, toss the broccoli in the oil and salt & pepper then place it into the casserole dish. Cook in the preheated oven for 7-10 minutes so that it is very lightly roasted then remove from the oven and allow to cool for a few minutes.
Add the broccoli to the bowl, mix together and then pour the mixture into the greased casserole dish. Sprinkle with remaining cheese and then bake for 40 minutes. Turn the oven to broil for 2-3 minutes so that the cheese is golden and bubbly then allow to cool for a few minutes before serving.