Begin by steaming the broccoli and cauliflower florets until they are tender yet firm, which usually takes 15-18 minutes. Use a large pot with about an inch of water, sprinkle with a little salt, and cover while cooking. Tender veggies ensure a smoother, more cohesive casserole. Strain and set aside to cool.
Heat olive oil in the same pot. Toss in the diced onion and cook until translucent, about 2 minutes. Add the baby spinach and minced garlic, stirring frequently. Cook until the spinach wilts down fully. Remove from heat to allow the mixture to cool slightly.
Whisk together the eggs, Greek yogurt, and milk in a large mixing bowl. This creates a creamy base for the casserole. Stir in 1 cup of shredded cheese, parsley, basil, chili flakes (if using), salt, and black pepper. Add the shredded rotisserie chicken, followed by the sautéed veggies and steamed cauliflower and broccoli. Mix it all to combine evenly.
Transfer the mixture into a greased 9x13 baking dish, spreading it out evenly. Sprinkle the remaining cheddar cheese over the top, followed by crushed parmesan crisps for a crunchy, golden topping.
Bake the casserole in a preheated oven at 375°F for about 35 minutes, or until the topping is lightly browned and bubbly. For an extra crispy top, broil for an additional 2-3 minutes. Cool slightly before serving for easy slicing.