Melt 3 tablespoons of butter in a large skillet over medium-high heat. Add the squash and onion. Season lightly with the salt and pepper. Cook until the squash is tender, about 10 minutes, stirring occasionally. Cover with a lid for a faster cooking time.
Remove from the heat and drain the squash and onion well using a colander, for a few minutes.
Meanwhile, in a bowl whisk together the eggs, mayonnaise, milk, and cheeses.
Lightly stir the squash mixture and wet mixture together. Scrape into a 12-inch buttered Dutch oven or 9 x 13-inch casserole dish.
Sprinkle the top with the crumbled crackers followed by the Parmesan cheese. Evenly pour the remaining 3 tablespoons of butter (melted) on top.
Bake until the casserole sets up and is lightly browned on top. For inside baking: 350 degrees F for about 25 minutes. Serve warm.
Notes
Tip: if you using Ritz crackers for the topping, use about 2 sleeves.