Heat a grill pan over medium-high heat. Pat the steak dry and brush lightly with olive oil. Season with 11/2 teaspoons salt and 2 to 3 teaspoons pepper. Cook, turning once, 10 to 13 minutes total for medium rare. Transfer the steak to a cutting board to rest, about 10 minutes.
Meanwhile, melt the butter in a medium saucepan over medium heat. Add the onion and cook until softened, about 5 minutes. Add the flour and cook, stirring, 1 minute. Remove from the heat and gradually whisk in the milk. Return to medium heat and simmer, whisking, until slightly thickened, about 2 minutes.
Stir in the spinach, 1/4 cup water, 1/2 cup parmesan, the lemon zest and nutmeg and cook, stirring occasionally, until the sauce thickens and the spinach is tender, about 10 minutes. Thin with water, if needed. Stir in the remaining 1/4 cup parmesan and salt to taste.
Thinly slice the steak and serve with the spinach.