Cook the lasagna noodles in a large pot of boiling salted water. Add 1 tablespoon olive oil to the water to prevent noodles from sticking. Cook noodles for about 10 minutes, drain and set aside on a baking sheet. Do not overcook noodles as they will cook more in the oven.
Prepare the Bechamel Sauce:
In a saucepan, melt the butter over medium heat. Add the flour and whisk until smooth, about 2 minutes. Gradually add the milk while continuing to whisk, until the sauce is smooth. Simmer for about 10 minutes until the sauce is thick enough to coat the back of a spoon. Stir in the salt, pepper, and nutmeg. Set aside.
Prepare the Spinach Filling:
Heat 1 tablespoon of oil in a large saute pan over medium heat, add half of the spinach, stirring until wilted and tender, about 3 minutes. Remove the spinach, place in a colander and squeeze out excess liquid, and then roughly chop. Repeat the process with the remaining spinach.
Heat the remaining olive oil and 1 tablespoon of butter in the saucepan and add the garlic. Saute until fragrant, about 1 to 2 minutes, then add scallions. Saute another 2 to 3 minutes until scallions are soft, then add the spinach along with the parsley, nutmeg, salt, and pepper. Remove from heat and allow to cool.
Stir the feta cheese and beaten egg into the cooled spinach mixture.
Assemble the lasagna:
Coat the bottom of a 13 by 9-inch lasagna pan with a thin layer of bechamel sauce. Arrange 4 noodles lengthwise on the sauce, slightly overlapping each other. Pour 1/3 of the bechamel sauce over the noodles. Then add 1/3 of the spinach mixture, spreading it out evenly. Top the spinach mixture with 1/4 cup of grated Parmesan. Repeat with 2 more layers of noodles, sauce, spinach mixture, and Parmesan.
For the topping, unroll the phyllo dough on a flat surface and keep it covered with a damp towel so that it doesn't dry out. Place a sheet of phyllo on top of the lasagna. Brush with melted butter and sprinkle some bread crumbs evenly. Top with another phyllo sheet, brush with more butter and sprinkle with more bread crumbs. Repeat with the 2 remaining layers of phyllo. Make sure the edges are nicely sealed with butter so that they don't curl up during the baking process. Top the final layer with a mixture of the remaining melted butter, bread crumbs, and Parmesan.
Bake in the preheated oven for 25 to 30 minutes, being careful not to overcook the phyllo. Remove from oven and let cool slightly before serving. Cut into squares and serve.