Cut off and discard stem end and green top from leek and remove outer layer. Slice leek in half lengthwise and run each half under cold running water, separating layers to fully clean. Chop into large pieces.
Wash and drain chard. Cut off discolored stems. Cut chard lengthwise and chop into large pieces.
In large saucepan, combine broth, leek, and chard. Bring to boil over high heat. Cover, lower heat, and simmer 12 minutes or until leek and chard are very soft.
Using slotted spoon, transfer leek and chard to blender or food processor. Add 1/2 cup cooking liquid and purée until smooth. Transfer puréed vegetable back into pot. Add milk and pepper, stir constantly. Cook 3 to 4 minutes, being careful not to scald milk.
Ladle soup into bowls. Garnish with chopped egg, chives, and salt to taste.