Preheat oven to 400 degrees.
Line a large baking sheet with parchment paper.
Place the eggplant, sweet potatoes, zucchini, onion and bell pepper on the sheet.
Mix with thyme leaves, drizzle with olive oil, pinch of salt, and pepper flakes.
Roast until softened, about 30 minutes. Reduce oven temperature to 350 degrees.
Line the base and sides of a deep, 10-inch baking pan with parchment paper. Layer the roasted vegetables in the pan, alternating between each.
Whisk together the eggs, 1/4 cup oil, and yogurt. Season with salt and pepper.
Pour over the vegetables in the pan, add the broccolini and sprinkle feta cheese over the top.
Bake until the middle is just set, 50-60 minutes.
Cool for 10 minutes and slice into individual servings.