Preheat oven to 350 degrees.
In a large skillet heat olive oil and saute onions and garlic just until they start to soften.
Add mushrooms and ¼ teaspoon salt and cook the mushrooms down until they have released most of their water, about 3 minutes.
Stir in squashes and quinoa. Continue to cook, stirring constantly until the quinoa is slightly toasted- about 4 minutes.
Add 3 C of chicken broth to the quinoa and bring to a simmer. Cover, turn heat to low and cook for 15-20 minutes or until most of the liquid is absorbed.
Transfer quinoa mixture to a greased 9x13" pan. Set aside.
Combine milk, eggs, salt and pepper in a separate container and whisk until smooth. Stir in shredded cheese.
Pour egg mixture over quinoa and slightly stir to combine. Mixture will be soupy.
Combine panko bread crumbs with melted butter, parsley, and remaining cheese. Sprinkle over the top of the quinoa.
Bake for 35-45 minutes or until the center is set and top is golden.
Let cool for at least 10 minutes before serving.