Preheat oven to 400°F (204°C). Lightly coat an 8-inch or 9-inch ceramic baking dish with extra virgin olive oil.
Place thawed spinach in a clean kitchen towel and squeeze firmly over the sink until no more liquid drips out. Set aside.
Warm 2 tablespoons of extra virgin olive oil in a large skillet over medium heat. Add diced onion and cook for 4–5 minutes until soft. Add minced garlic and cook for 1 minute until fragrant.
Sprinkle flour over the onions and garlic, stirring continuously for 1–2 minutes. Gradually whisk in the milk to avoid lumps. Bring to a gentle simmer for 3–5 minutes until thickened. Remove from heat.
In a large bowl, combine the warm milk mixture, drained spinach, ricotta cheese, beaten eggs, ground nutmeg, salt, black pepper, ¾ cup mozzarella, and ¼ cup Parmigiano-Reggiano. Mix until uniform.
Transfer mixture to prepared baking dish, smoothing the top with a spatula.
Sprinkle remaining ¾ cup mozzarella and ¼ cup Parmigiano-Reggiano evenly over the surface.
Bake uncovered for 25–30 minutes until edges are bubbling. Switch oven to broil for 2–3 minutes until top is golden brown and blistered.
Rest for 5–10 minutes before serving.