Preheat the oven to 450F or 425F convection. Line a baking sheet with parchment paper.
Add the squash, mushrooms, oil, thyme, and ½ teaspoon salt and use your hands to toss to coat. Spread out into an even layer and bake for 20-25 minutes or until the squash is tender and browning.
Add the greens and use a spatula to toss everything together so that the greens get lightly coated in the oil, then spread everything out again and bake for 2-3 minutes. Remove from the oven and reduce the oven temperature to 300F. Use a spatula to toss everything together again and spread out to evenly distribute the veggies.
In a large bowl, whisk together the eggs and remaining ½ teaspoon salt. Pour slowly all over the veggies. Top with goat cheese, a sprinkling of parmesan (if using), and black pepper.
Bake for 10-20 more minutes or until eggs are just cooked through (see recipe note, times will vary based on oven temps, so just use your judgment).
Let sit for 5-10 minutes before slicing so that it can cool down and continue to cook through from the residual heat.
Use a knife to slice it up and a fish spatula to carefully get the slices out of the pan.