Skillet Mushroom Lasagna

Ingredients

• extra-virgin olive oil, divided
• mixed mushrooms , trimmed, torn into 1" pieces
• tsp. kosher salt, plus more
• thyme leaves
• large shallot, finely chopped
• garlic cloves, finely grated
• all-purpose flour
• whole milk
• Freshly ground black pepper
• lemon
• crème fraîche or thinned sour cream
• regular lasagna noodles, broken in half
• mozzarella, thinly sliced
• Finely grated Parmesan