Make the béchamel: In a medium, heavy saucepan melt the butter over medium heat until bubbling. If using olive oil, simply heat it over medium flame. Add the flour. Stir with a wire whisk until the butter-flour mixture (roux) starts to turn light tan, about 4 minutes on medium-low heat. Add the milk in a slow, steady stream, whisking all the while. Continue to stir the mixture over medium heat with a wire whisk until the béchamel thickens to the consistency of a loose batter. Season to taste with salt, pepper and nutmeg. Quickly break and stir in the egg, whisking it vigorously. Set aside, covered.
In a large skillet over medium flame heat two tablespoons of olive oil and cook the onions until lightly golden. Stir in the chopped garlic to soften. Transfer the mixture to a mixing bowl.
In the same skillet, add the chopped spinach and heat until most of the liquid it exudes cooks off. Drain the spinach in a colander and add to the onion mixture. Don’t worry about draining the spinach to perfection! A little bit of excess liquid will help the lasagna strips cook better when baking.
Mix in the herbs, feta and half the grated cheese. Season to taste with salt, pepper and nutmeg. Stir 1 1.2 cups of the béchamel into the spinach mixture.
Preheat the oven to 350F / 170C. Lightly oil or butter an 9 X 13-inch baking pan. Spread 2 – 3 tablespoons of the béchamel mixture over the bottom of the pan. Place 3 – 4 lasagna strips on the bottom. Drizzle them with a little of the remaining olive oil. Spread one third of the spinach mixture over the strips. Drizzle in a spoonful or two of the béchamel. Place another 3 – 4 lasagna strips over the spinach mixture and repeat the process. Repeat once again, to have three layers of lasagna strips and finish with the spinach mixture. Ladle in the remaining béchamel, spreading it evenly over the surface. Sprinkle with remaining grated cheese.
Bake for about 45 minutes, until the lasagna strips are soft, the béchamel set and the surface lightly golden. Remove, let cool slightly and serve.