Step 1: Preheat your oven to 375°F (190°C). Lightly butter a 9x13 inch baking dish and set aside.
Step 2: In a large skillet or Dutch oven, melt 1/4 cup butter over medium heat. Add the chopped yellow onion and cook until softened and translucent, about 5-7 minutes. Add the thinly sliced yellow squash to the skillet and cook, stirring occasionally, for 8-10 minutes, or until tender-crisp. Remove from heat.
Step 3: In a separate medium saucepan, melt 3 tablespoons of butter over medium heat. Whisk in the flour and cook for 1-2 minutes, stirring constantly, to create a roux.
Step 4: Gradually whisk in the milk, a little at a time, until smooth and no lumps remain. Continue to cook, whisking constantly, until the sauce thickens, about 5-7 minutes. Remove from heat.
Step 5: Stir 1 cup of grated Parmesan cheese into the thickened milk sauce until melted and smooth. In a small bowl, lightly beat the egg, then temper it by whisking a small amount of the hot cheese sauce into the egg, then slowly pour the egg mixture back into the main cheese sauce, whisking constantly. Season with salt and pepper to taste.
Step 6: Add the cooked yellow squash and onions to the cheese sauce, stirring gently to combine. Pour the mixture into the prepared baking dish.
Step 7: In a small bowl, combine the crushed Ritz crackers with 1-2 tablespoons of melted butter (optional, but adds extra crispness and flavor). Sprinkle the buttered cracker crumbs evenly over the top of the squash mixture in the baking dish.
Step 8: Bake for 30-35 minutes, or until the casserole is bubbly around the edges and the cracker topping is golden brown and crispy. Remove from the oven and let rest for 5-10 minutes before serving. Enjoy your delicious creamy Parmesan yellow squash casserole!