Trim and discard stem end and tough green top from leek; peel off and discard outer layer. Cut leek in half lengthwise and hold each half under cold running water, separating layers to rinse well. Coarsely chop.
Rinse and drain chard. Trim off discolored stem ends and discard. Thinly slice stems crosswise and coarsely chop leaves.
In a 3- to 4-quart pan, mix broth with leek and chard stems and leaves. Bring to a boil over high heat; cover, reduce heat, and simmer until vegetables are very tender when pierced, about 12 minutes.
With a slotted spoon, transfer vegetables to a blender or food processor. Add about 1/2 cup cooking liquid and whirl until smooth. Pour purée back into pan. Add milk and 1/4 teaspoon pepper; stir just until steaming, 3 to 4 minutes.
Ladle bisque into wide bowls. Sprinkle equally with chopped egg and chives. Add salt and more pepper to taste.