In a large mixing bowl, beat 3/4 cup butter until fluffy.
Add both sugars and beat well.
Add egg, vanilla, and salt and beat again.
Add the flour and beat until well mixed.
Cover and chill for 1 hour.
Preheat oven for 350°.
Shape the dough into 1-inch balls.
Place on an ungreased cookie sheet 2 inches apart.
Press down in the center with your thumb (step 2).
Bake for about 12 minutes.
Cool on wired racks.