Heat oil and butter in a large skillet over medium heat until the butter melts. Add the onions and cook, stirring occasionally, until softened and translucent, about 10 minutes. Add the flour and cook for 2 minutes. Stir in the half-and-half and cook, stirring occasionally, until thickened, about 10 minutes.
Squeeze as much liquid as possible out of the spinach and add to the sauce. Add the cheeses and cook, stirring, until melted. Add the nutmeg and season to taste with salt and pepper. Serve hot.