Make the salad dressing by combining the first 6 ingredients in a salad bowl. Set aside until needed.
Set the oven rack 6” beneath the broiler. Preheat the broiler. Place the eggs and milk in a second bowl and beat until the egg whites and yolks are well blended. Mix in the ham, nutmeg, salt and pepper. Coat a 9” or 10” non-stick skillet with vegetable oil spray and set over medium heat. Add the egg mixture and cook, gently stirring and scraping to move uncooked portions of egg to the bottom of the pan, until the eggs are just set, but still quite moist. Remove the pan from the heat and top eggs with pieces of Brie. Broil the frittata until the top is light golden and the cheese is just melted.
Let frittata set a few minutes before carefully removing from the pan and sliding onto a cutting board. Toss the salad greens into the dressing in the salad bowl. Cut the frittata into quarters and place one on each of 4 plates. Top each frittata with salad greens. Decorate the top of the greens with cherry tomatoes, carrots and cucumber and serve.