In a saucepan combine the potatoes, peeled and cut into 1-inch pieces, with enough cold water to cover them by 1 inch and simmer them for 10 to 15 minutes, or until they are tender.
While the potatoes are cooking, in a heavy skillet cook the leek, the onion, and the garlic in 2 tablespoons of the butter, covered, over moderate heat, stirring occasionally, for 10 minutes, or until the vegetables are soft.
Drain the potatoes, return them to the pan. Mashed the potatoes. Stir in the milk, the remaining 3 tablespoon butter, the leek mixture, the goat cheese, the chives, and salt and pepper to taste.