To make the filling, put the chocolate in the top of a double boiler and set over, not in, barely simmering water. Melt the chocolate, smoothing it with a whisk.
Remove the insert and let the chocolate cool, stirring occasionally. Set aside.
Using an electric mixer, beat the butter in a large bowl until creamy.
Gradually add the sugar and continue to beat, scraping down the bowl as needed, until the mixture is light-textured, about 5 minutes.
Scrape the melted chocolate into the butter mixture. Beat for 30 seconds.
Add the eggs, one at a time, beating on medium-high speed for 4 minutes after each addition. The filling should be very light and creamy.
Add the vanilla and blend for 10 seconds to incorporate.
Scrape the filling into the chilled pie shell and smooth the top with a spoon.
Cover with loosely tented aluminum foil and refrigerate for at least 3 hours or overnight.