1 1/2 pounds Portobello mushrooms, stems discarded and caps halved and sliced
1 1/2 pounds cremini mushrooms, trimmed and sliced
1/4 cup olive oil
1 tablespoon fresh thyme leaves
kosher salt and black pepper
1/2 cup (1 stick) unsalted butter, plus more for the foil
1/2 cup all-purpose flour
6 cups whole milk
1/4 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
Heat oven to 400° F. Dividing evenly, on 2 rimmed baking sheets, toss the mushrooms with the oil, thyme, 3/4 teaspoon salt, and 1/4 teaspoon pepper. Roast, tossing once, until tender, 20 to 25 minutes; let cool. Reduce oven to 375° F.
Meanwhile, melt the butter in a large saucepan over medium heat. Add the flour and cook, stirring, until foamy, 1 to 2 minutes (do not let the mixture darken). Slowly whisk in the milk. Bring to a simmer and cook, whisking often, until thickened, 3 to 5 minutes. Remove from heat, add the nutmeg, 1/2 cup of the Parmesan, 3/4 teaspoon salt, and 1/4 teaspoon pepper, and stir to combine.
Spread 1/2 cup of the milk mixture in the bottom of a 9-by-13-inch baking dish. Top with 4 noodles (breaking to fit, if necessary), 1/2 cup of the remaining milk mixture, 1 package of the spinach, and half the mushrooms; repeat. Top with the remaining 4 noodles, milk mixture, and 1/2 cup of Parmesan.
Cover the dish with buttered foil and bake until the filling is bubbling, 15 to 20 minutes. Uncover and bake until the noodles are tender and the top is brown, 20 to 25 minutes more. Let rest for 15 minutes before serving.