Preheat your oven to 350°F and lightly grease a 9x13 inch baking dish with cooking spray. While the oven heats, prepare all your ingredients: slice the yellow squash into 1/4-inch thick rounds, seed and finely mince the chiles, chop the scallions (separating white and green parts for different timing), and mince the garlic. Soften the cream cheese to room temperature if not already done. Having everything ready will make the cooking process smooth and prevent overcooking any components.
Heat the butter and oil in a large skillet over medium-high heat until the butter is foaming. Add the sliced squash in a single layer and cook for 4-5 minutes until it begins to soften and develop light golden edges, stirring occasionally. Add the minced chiles and cook for another 3-4 minutes to soften them and release their flavor. Then stir in the white parts of the scallions, garlic, smoked paprika, salt, and pepper, cooking for about 1 minute until fragrant. I like to add a generous pinch of salt at this stage because it helps draw out the squash's natural moisture and concentrates the flavors.
In a medium bowl, whisk together the mayonnaise, softened cream cheese, eggs, and 1/2 cup of Parmesan until smooth and well combined. The mixture should be creamy and homogeneous with no lumps of cream cheese visible. This mixture will act as a rich custard-like binder that holds everything together while baking.
Spread half of the creamy mixture from Step 3 evenly across the bottom of your prepared baking dish. Layer the sautéed squash mixture from Step 2 over the cream base, then pour the remaining half of the creamy mixture over the top, spreading it gently to cover all the squash. This layering technique ensures even cooking and helps distribute the creamy sauce throughout the dish. Sprinkle the remaining 1/4 cup of Parmesan evenly over the top, then finish with the crushed crackers for a crunchy texture contrast.
Bake in the preheated 350°F oven for 30-40 minutes until the top is golden brown and the filling is set but still slightly moist in the center (it will continue to set as it cools). The casserole should jiggle just slightly when you gently shake the dish—this means the custard is perfectly set without being overcooked. Remove from the oven and let rest for 5 minutes before serving. I like to scatter the reserved green parts of the scallions over the top just before serving for a fresh pop of color and mild onion flavor that contrasts beautifully with the rich, creamy interior.