Clean and slice the white part and 2 inches of the green part of the leeks.
In a lidded frying pan, saute the leeks in the butter until wilted. Add balsamic vinegar, salt and pepper; cover and cook on low heat until leeks are tender, about 30 minutes. Remove lid and increase heat to moderately high; add cream and cook, stirring, until cream is absorbed, about 1 minute. Remove pan from heat and cool slightly.
Spread the goat cheese evenly over the bottom of the pie shell. Spoon the leek mixture evenly over the cheese. Bake in a 350°F oven until heated through and browned on top, about 15 minutes.