Combine apricots and water in a heavy saucepan; cook over medium heat 30 minutes or until tender. Drain; place apricots in a medium mixing bowl, and mash until smooth. Stir in 3/4 cup sugar; chill overnight.
Cream butter; gradually stir in flour and cottage cheese. Shape mixture into 1-inch balls, and chill overnight.
Combine remaining sugar and pecans in a small mixing bowl; set aside.
Roll balls to a 3-inch diameter on a lightly floured surface. Place 1 teaspoon chilled apricot mixture in center; fold over, and pinch edges together. Dip into egg whites; roll in reserved pecan mixture, and place on greased cookie sheets. Bake at 375° for 12 to 14 minutes. Remove from cookie sheets, and cool on wire racks.
Note: Cottage cheese balls and apricot mixture may be refrigerated up to 2 weeks to prepare turnovers as needed.