Heat the oven to 350°F. Coat a 9x13-inch baking dish lightly with cooking spray.
Process 2 large eggs, 1 1/2 cups half-and-half, 1 1/4 teaspoons kosher salt, and 1/4 teaspoon black pepper in a blender or food processor until smooth, about 10 seconds. Add 1 cup of the corn kernels and pulse until the corn is very finely chopped, 5 to 10 (1-second) pulses.
Transfer to the baking dish. Stir in the remaining 4 cups corn kernels and 3/4 cup of the shredded cheddar cheese. Spread into an even layer. Sprinkle with the remaining 3/4 cup shredded medium cheddar cheese.
Stir 40 coarsely crushed butter crackers and 4 tablespoons melted unsalted butter together in a medium bowl until evenly moistened. Sprinkle evenly over the top of the casserole.
Bake until the topping is golden, the corn is heated through, and the casserole is set in the middle, 25 to 30 minutes. Let cool for 5 minutes before serving.