8 tablespoons (1 stick) unsalted butter, plus more for greasing baking dish, divided
3 medium cloves garlic, minced (about 1 tablespoon)
2 pounds washed fresh spinach leaves, roughly chopped
1 pint (2 cups) heavy cream, divided
1/2 teaspoon freshly grated nutmeg
Kosher salt and freshly ground black pepper
1 pound cottage cheese
1/2 cup chopped fresh parsley leaves, divided
1 egg
1 package (15 sheets) no-boil lasagna noodles
24 ounces button, cremini, or shiitake mushrooms, stems removed, finely chopped (see note above)
2 medium shallots, finely chopped (about 1/2 cup)
2 teaspoons minced fresh thyme leaves
1 tablespoon soy sauce
2 teaspoons fresh juice from 1 lemon
2 tablespoons flour
2 cups whole milk
12 ounces whole milk mozzarella, grated, divided
2 ounces Parmiggiano-reggiano, grated