Mushroom and Spinach Lasagna

Name: Mushroom and Spinach Lasagna
Ingredients: 1 ½ pounds Portobello mushrooms, stems discarded and caps halved and sliced; 1 ½ pounds cremini mushrooms, trimmed and sliced; ¼ cup olive oil; 1 tablespoon fresh thyme leaves; kosher salt and black pepper; ½ cup (1 stick) unsalted butter, plus more for the foil; ½ cup all-purpose flour; 6 cups whole milk; ¼ teaspoon ground nutmeg; 1 cup grated Parmesan (4 ounces); 12 no-boil lasagna noodles; 2 10-ounce packages frozen chopped spinach, thawed and squeezed dry