Fill a large bowl with ice water; set aside.

Bring 12 cups water and 2 tablespoons salt to a boil in a large stockpot over high.

Add spinach; cook, stirring constantly, until wilted, about 1 minute.

Using a slotted spoon or a spider, transfer spinach to prepared ice water; let cool 1 minute.

Drain well; place spinach on a clean kitchen towel and squeeze firmly over sink to remove as much liquid as possible.

Place spinach in a food processor; pulse until roughly chopped, 4 to 6 pulses. Set aside.

Preheat oven to 350°F with rack about 10 inches from upper oven heat source.

Meanwhile, melt butter in a 10-inch cast-iron skillet over medium-high.

Add onion; cook, stirring occasionally, until tender, about 4 minutes.

Add garlic; cook, stirring constantly, until fragrant, about 30 seconds.

Reduce heat to medium.

Stir in flour, mustard, cayenne, nutmeg, and remaining 1 teaspoon salt; cook, stirring constantly, 1 minute.

Gradually whisk in milk until combined; bring mixture to a boil over medium, whisking occasionally.

Boil, whisking occasionally, until thickened, about 2 minutes.

Stir in cheddar and ½ cup of the Monterey Jack cheese; cook, stirring constantly, until smooth, about 1 minute.

Remove from heat; stir in reserved spinach until combined.

Sprinkle evenly with remaining 1 cup Monterey Jack cheese.

Bake in preheated oven until bubbly, about 15 minutes.

Increase oven temperature to broil (do not remove skillet from oven).

Broil until browned, 2 to 5 minutes.

Let stand at room temperature 5 minutes before serving.