Heat milk in a pot with basil, cilantro, and tarragon; bring to a boil. Take pot off of heat, place lid on pot, and let rest 10 minutes. Strain and dispose of herbs.
Whisk eggs, egg whites, and granulated sugar 5 minutes or until mixture is pale yellow. Slowly whisk milk mixture into egg mixture. Spoon mixture into 8 custard cups.
Arrange cups in a roasting pan and place in the middle rack of an oven; pour 2 inches of water in pan. Bake at 350 degrees F for 20 minutes or until a knife comes out clean. Take cups out of roasting pan and let cool to room temperature. Chill 2 to 4 hours.
Reduce oven temperature to 150 degrees F. Sprinkle brown sugar in a baking dish and bake 10 minutes or until moisture is evaporated. Do not melt sugar.
Switch oven to broil. Top each custard cup with brown sugar. Arrange custard cups on a cookie sheet and broil 2 to 3 minutes or until sugar is caramelized.