Combine all marinade ingredients in a shallow dish and place mushrooms in it with gills down. Let them sit for about 10 minutes, flipping them over and let sit another 10 minutes.
Preheat oven to 400 degrees F.
Slice delicata squash in half lengthwise and remove seeds. Place face down on cutting board and slice into ¼-inch pieces. Drizzle with olive oil and squash seasonings, then bake on small sheet pan for 8 to 10 minutes. Set aside.
Drizzle 1 tablespoon of olive oil into large skillet over medium-high heat. Cut marinated mushrooms into ¼ -inch slices and sauté. until soft with slight browning. Set aside.
Assemble flatbreads in this order: crumbled feta, squash, mushrooms, and red onions. Drizzle with olive oil, to taste and bake for 8 to 10 minutes until edges look crisp. Garnish with fresh thyme, pomegranate seeds, and balsamic glaze. Slice each into 4 to 5 pieces. Serve immediately.