Half the delicata squash, lengthwise, and scrape out the seeds and stringy stuff. Dice the squash and set aside.
Preheat the oven to 400 degrees F.
In a small sauté pan, set over medium heat, pour in the olive oil. when warm, add the onion and delicata squash, along with a few pinches of salt. Add the leaves of one sprig of thyme and a few turns of freshly cracked pepper, along with a pinch of red pepper flakes. Cook for 5 to 6 minutes, until the squash is cooked all the way through. Give it a taste and cook for a few minutes longer.
Meanwhile, to the bowl with the caramelized onions, add the brie, 1 cup of mozzarella, the creme fraiche, leaves from a sprig of thyme, a pinch of crushed red peppers, a pinch of salt and a few rounds of freshly ground pepper. Mix and mash it together until combined.
Transfer half of it to a 6 x 6-inch casserole dish or something comparable, and then top it with a handful of mozzarella and a few spoonfuls of squash. Repeat until you’ve worked your way through all of the cheese and squash. Top with a generous handful of cheese.