Melt the butter in a medium pot over medium heat. Sprinkle the flour over the melted butter, whisk to combine, and cook for about 1 minute. Slowly whisk the milk into the butter-flour mixture, season with salt, pepper, and nutmeg, and cook for a couple of minutes to thicken. Stir in half the Parmigiano (about 3/4 cup). Set the sauce aside.
Heat the EVOO in a large cast-iron skillet over medium-high heat. Toss in the cracked clove of garlic and cook for a minute or two to release its aroma and flavor. Add the chard and cook, turning occasionally to wilt down, 2 minutes. Add the spinach a few handfuls at a time, wilting each batch before adding more. Season the greens with salt and pepper.
Break the pasta sheets into large pieces. Toss them into the pan with the greens and give them a good stir to incorporate them. Pour the sauce into the skillet, stirring again to incorporate everything evenly. Smooth the top down and sprinkle the remaining 3/4 cup Parmigiano into the skillet. Cover the skillet with foil, transfer to the oven, and bake the lasagna for 30 minutes. Remove the foil from the pan and let it finish cooking uncovered for about 15 more minutes to brown the cheese.