Preheat your oven to 375°F (190°C). Grease a 9x13 inch baking dish thoroughly with butter.

Drain the canned whole green chiles and carefully pat them very dry with paper towels. This helps prevent the casserole from becoming watery. You can leave them whole or carefully slice them lengthwise to lay flatter.

In a large mixing bowl, whisk together the large eggs, whole milk, all-purpose flour, salt, and black pepper until the mixture is smooth and well combined, with no lumps of flour.

Arrange half of the drained green chiles in a single layer at the bottom of the prepared baking dish.

Sprinkle 1.5 cups of the shredded Monterey Jack cheese evenly over the layer of chiles.

Place the remaining half of the chiles over the cheese layer.

Carefully pour the egg custard mixture evenly over the chiles and cheese. Gently tilt the dish to ensure the custard fills all gaps.

Top the casserole with the remaining 1.5 cups of shredded Monterey Jack cheese.

Bake for 30 minutes, covered with aluminum foil. Then, remove the foil and bake for an additional 15-20 minutes, or until the casserole is golden brown on top, the cheese is bubbly, and a knife inserted into the center comes out clean.

Once baked, remove the casserole from the oven and let it rest for 5-10 minutes before serving. This allows the layers to set and makes for easier slicing. Garnish with fresh chopped cilantro, if desired.