Stovetop Mushroom Lasagna

Name: Stovetop Mushroom Lasagna
Ingredients: 5 Tbsp. extra-virgin olive oil, divided; 12 oz. mixed mushrooms (such as maitake, oyster, shiitake, and/or crimini), trimmed, torn into 1 pieces; 1½ tsp. tsp. kosher salt, plus more; 2 Tbsp. thyme leaves; 1 large shallot, finely chopped; 3 garlic cloves, finely grated; 2 Tbsp. all-purpose flour; 2¾ cups whole milk; Freshly ground black pepper; 1 lemon; ⅓ cup crème fraîche or thinned sour cream; 8 oz. regular lasagna noodles, broken in half; 4 oz. mozzarella, thinly sliced; Finely grated Parmesan (for serving)