Place an oven rack in the center position. Preheat the oven to 375°F. Spray a medium ovenproof baking dish with nonstick cooking spray. Place the breasts in the prepared pan. Season with salt and pepper. Set aside.
Place the onion in a microwave safe bowl. Add a splash of water and microwave until clear and translucent, about 30 seconds. Set aside to cool.
In a mixing bowl, combine the spinach, artichokes, garlic, skyr, mayonnaise, mustard, and reserved onion. Spoon the mixture on top of the chicken and spread to an even thickness. Sprinkle over the cheese.
Transfer to the oven and bake until the chicken reaches 165°F when tested with an instant read thermometer, 30 to 35 minutes. Serve immediately.