In the bowl of an electric mixer fitted with a paddle attachment, beat butter and Sugar until light and fluffy.
With mixer on low speed, add flour and egg in two parts and beat just until combined.
Shape dough into a disk, wrap in waxed paper, and chill 1 to 24 hours.
Roll out dough on a lightly floured surface with a lightly floured rolling pin to 1/8- to 1/4-inch thickness. Transfer to a 9-inch tart pan, press in sides and bottom, and trim any excess dough around edges.
Preheat oven to 350 °F.
Cover dough with a large piece of foil and weigh down with uncooked rice or beans.
Bake 20 minutes, remove foil and beans, and bake 10 minutes longer, or until light golden brown.
Cool on a rack.
To make the filling, preheat oven to 375 °F.
In a medium saucepan over medium heat, bring cream, milk, and basil just to a boil.
Remove pan from heat, cover, and steep 10 minutes.