Preheat the oven to 350°F. Spray a 9×13-inch casserole dish with cooking spray or lightly oil it.
Crack 12 eggs into a large bowl. Add 1/2 cup milk, 1/4 cup melted (and slightly cooled) butter, 1/4 teaspoon salt, and 1/2 teaspoon ground black pepper.
Whisk the mixture until the eggs, milk, butter, salt, and pepper are well combined and uniform in color.
Pour the egg mixture into the prepared 9×13-inch dish and spread it evenly.
Bake on the middle rack for 12 minutes.
Remove the dish from the oven and use a fork to run through the eggs, breaking them up into curds and stirring gently to redistribute any liquid.
Return the dish to the oven and bake for another 8 minutes.
Remove from the oven and use a fork to stir/scramble any still-wet egg into the curds; let the dish rest 1–2 minutes so residual heat finishes cooking. Serve.