Preheat the oven to 350°F. Butter a 9x13-inch baking dish.
Melt the butter in a large skillet over medium heat. Add the diced onion and cook 4–5 minutes until soft. Add the garlic and cook 1 minute more.
Add the squeezed spinach to the skillet and stir to combine with the onion and garlic. Cook 2 minutes to evaporate any remaining moisture, then remove from heat and let cool slightly.
In a large mixing bowl, combine the cooked rice, spinach mixture, cream cheese, 1 cup Swiss, and Parmesan. Stir until the cream cheese is fully incorporated. Season with nutmeg, salt, and pepper.
In a small bowl, whisk the eggs and milk together until smooth. Fold into the rice mixture until fully combined, with no visible streaks of egg.
Transfer the mixture to the prepared baking dish and spread into an even layer.
Toss the panko with the melted butter, then sprinkle evenly over the top along with the remaining ¼ cup Swiss.
Bake for 35–40 minutes, until the center is set and the top is golden. Broil 1–2 minutes if extra color is needed, watching closely.
Rest 10 minutes before cutting into squares and serving.