Heat oven to 350°F (325°F for dark or nonstick cookie sheet). In medium bowl, beat 1 cup butter, 3/4 cup powdered sugar and the vanilla with electric mixer on medium speed until smooth. Beat in flour until well blended. Stir in finely chopped apricots.
Shape dough into 1-inch balls. On ungreased cookie sheet, place balls 2 inches apart.
Bake 8 to 12 minutes or until bottoms are light golden brown. Cool 1 minute; remove from cookie sheet to cooling rack. Cool completely, about 30 minutes.
In small bowl, beat 1/4 cup butter and the apricot preserves with electric mixer on medium speed until smooth. Beat in 1 cup powdered sugar until well blended. Spread frosting over cookies. Top each cookie with apricot strip.