Preheat the oven to 375 degrees F. Lightly coat a 12-cup nonstick muffin pan with nonstick cooking spray.
Mist a large nonstick skillet with more nonstick cooking spray and put over medium heat. Add the turkey sausage and cook turning frequently until crisp, about 8 minutes. Transfer to a cutting board and coarsely chop.
In the same skillet, heat the oil. Add the mushrooms, shallot, and season to taste with salt and pepper. Cook until shallots are soft and the mushrooms are dry and lightly browned, about 10 minutes. Remove from the heat and stir in the baby spinach and nutmeg until the spinach is just wilted, but still bright green, 2 minutes. Set aside to cool slightly.
Meanwhile, whisk the eggs, egg whites, milk, and salt and pepper until well combined. Stir in the cheese. Divide the egg mixture evenly between the muffin tins, filling them about halfway. Top evenly with the mushroom and spinach mixture.
Bake until the quiches are well risen, golden brown and set, 20 to 25 minutes. Cool in the pan 5 minutes and then transfer to a wire rack.