Preheat oven to 375F.
Place a medium pot over medium heat and melt the butter. Sprinkle the flour over the melted butter and cook for about a minute. Slowly whisk the milk into the butter-flour mixture and season with salt, pepper and nutmeg. Let it thicken a couple of minutes then stir in about a cup of the Parmigiano. Set the sauce aside.
Place a large, cast iron skillet over medium-high heat with 2 turns of the pan of EVOO, about 2 tablespoons. Toss in the cracked clove of garlic and cook for a minute or two to release its aroma and flavor. Add in the chard and wilt it down, about 2 minutes, then wilt in the spinach a few handfuls at a time. Season the greens with salt and pepper.
Break the pasta sheets into large pieces. Toss them into the skillet and give them a good stir to incorporate them with the greens. Pour the sauce into the skillet, stirring again to evenly incorporate everything. Smooth the top down and sprinkle the remaining 1/2 cup of Parmigiano Reggiano over top of the lasagna.
Cover the skillet with foil and bake for 30 minutes. Remove the foil from the pan and let it finish uncovered for about 15 more minutes to brown the cheese.