Heat the olive oil in a 12-inch straight-sided skillet or cast iron pan (with a lid) over medium-high heat. Once the oil is hot, add the mushrooms and toss to coat. Let the mushrooms cook, undisturbed, until they begin to caramelize on the bottom, about 3 minutes. Give them a stir and cook until tender, about 2 minutes longer.
Add the shallot, garlic and thyme, and season with a pinch of salt and pepper. Cook, stirring, until the shallots and garlic are softened, about 1 minute. Add the wine and cook, stirring, until almost evaporated. Add 3 cups of the broth and bring to a simmer.
Pour the remaining ¾ cup of broth into a bowl and whisk in the cornstarch. Add the crème fraiche and whisk until smooth. Slowly pour the mixture into the skillet, whisking. Bring to a boil while whisking. Add 1 ½ teaspoons of salt and season generously with pepper.
Break each lasagna noodle into roughly 4 pieces and drop them into the sauce (it’s ok if there are smaller pieces). Once all the noodles have been added, give them a stir to coat, and then arrange the noodles in a flat layer (not all of the noodles will be covered). Reduce the heat to a simmer. Cover the skillet and cook 18-20 minutes, stirring twice during the cooking time, until the noodles are tender when pierced with a knife.
When the noodles are almost done, preheat the broiler with the rack in the upper third.
Once the noodles are tender, add the spinach by the handful and gently stir until wilted. Season to taste with salt and pepper. Sprinkle the fontina evenly over top of the lasagna, then dollop with the fresh ricotta. Finish with the Parmesan (I just grate it right over top, eyeballing). Grate some fresh nutmeg over top if you have it.
Place the pan in the oven under the broiler. Broil until the cheese is golden in spots (it takes about 5 minutes under my broiler, but every model is different). Let cool 5 minutes. Drizzle with truffle oil if desired, and serve. Do Ahead: The lasagna can be covered and stored in the fridge for up to 2 days. Reheat it in a 350˚F oven until warmed through.