Preheat your oven to 400F and place a large ovenproof skillet over medium-high heat. Add the oil and, when hot, follow with the mushrooms. Season with salt and pepper and cook for about five minutes or until lightly browned. Add the minced garlic and cook for an additional one to two minutes. Transfer the mushrooms and garlic to a plate and set aside.
Place the pan back over the heat and add the butter to melt. Sprinkle the flour on top of the butter and cook, whisking frequently, for about two minutes. Whisk in the milk and chicken broth and allow it to simmer, whisking occasionally until smooth and thick, about five minutes. Season with black pepper, a pinch of nutmeg, and some salt to taste.
Add the parsley, chicken, gnocchi, arugula, and cooked mushrooms and garlic and stir until everything is well combined. Sprinkle the top with the cheese and transfer the skillet to the oven. Bake for 20 – 25 minutes then turn on the broiler for about three minutes or until the top is lightly golden brown.
Remove the skillet from the oven and allow it to cool slightly before serving.