Step 1

Heat 3 Tbsp. oil in a large high-sided ovenproof skillet or small Dutch oven over medium-high. Add mushrooms and cook, undisturbed, until starting to brown, about 4 minutes. Season with salt and cook, tossing occasionally, until golden brown, 5–7 minutes. Transfer to a bowl; mix in thyme.

Step 2

Reduce heat to medium-low. Heat remaining 2 Tbsp. oil in skillet. Add shallot and garlic and cook, stirring, until starting to soften, about 1 minute. Sprinkle flour over and cook, stirring, until golden, about 1 minute.

Step 3

Add milk, pepper, 1½ tsp. salt, and 1¾ cups water, then finely grate zest of ¼ lemon into pan. Stir to dissolve flour, increase heat to medium, and cook until gently bubbling. Reduce heat to low, whisk in crème fraîche, and stir to combine. Add about one third of noodles, pushing down into sauce to submerge, followed by a third of  mushrooms. Repeat with half of remaining noodles and mushrooms. Top with another layer of noodles. Set remaining mushrooms aside. Cover with a lid or foil and cook 15 minutes. (If lasagna threatens to boil over, use very lowest heat and/or prop open the cover.) Uncover; cook, gently lifting and separating noodles occasionally with tongs or a spatula to let sauce flow around, until sauce is thickened and noodles are cooked through, 6–8 minutes. Mixture should be bubbling gently; adjust heat as necessary. Remove from heat.

Step 4

Heat broiler. Top lasagna with mozzarella and reserved mushrooms. Broil until cheese is bubbling and browned in spots, about 2 minutes. Finely grate more lemon zest over. Sprinkle with Parmesan; season with pepper. Let sit 5–10 minutes. Cut lemon into wedges and serve with lasagna.