Preheat the oven to 350°F. Spray an 8-inch or 9-inch square baking dish with nonstick cooking spray; set aside.
Melt 1 tablespoon of butter in a large skillet over medium heat. Add the zucchini, onion, and bell pepper to the skillet, cooking and stirring until the veggies are tender, about 10 minutes. Remove from the heat and season with salt and pepper to taste. You do not need to drain off any liquid from the vegetables.
Meanwhile, in a large mixing bowl, whisk together the eggs, cheese, sour cream, and mayonnaise. Add the zucchini mixture and the chicken; stir gently to combine. Season with about ¼ teaspoon of salt and ¼ teaspoon of pepper.
Transfer the mixture to the prepared baking dish.
In a separate bowl, combine the crushed crackers with 1 tablespoon of melted butter. Sprinkle the buttered cracker crumbs over the top of the casserole.
Bake until the casserole is set and the topping is golden brown, about 30-35 minutes. Garnish with chopped fresh parsley or basil, if desired.