Preheat the oven to 375 degrees F (190 degrees C). Grease a 9x13-inch baking dish.

Place chicken breasts in the prepared baking dish. Whisk together Asiago cheese, mayonnaise, Greek yogurt, wine, oregano, garlic powder, onion powder, and nutmeg in a small bowl. Pour evenly over chicken breasts.

Bake, uncovered, in the preheated oven until chicken is no longer pink in the center and the juices run clear, 40 to 50 minutes. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C). Top with black pepper, balsamic vinegar, and additional Asiago cheese just before serving.