Heat oven to 400°F. Lightly spray cookie sheet with nonstick cooking spray. Lightly spoon flour into measuring cup; level off. In medium bowl, combine 1 3/4 cups of the flour and powdered sugar. With pastry blender or fork, cut in butter until mixture resembles coarse crumbs.
Reserve 2 tablespoons of the apricots for topping; stir remaining apricots into flour mixture. In small bowl, combine 1/3 cup milk and eggs; beat well. Add to flour mixture; stir just until soft dough forms.
Sprinkle work surface with remaining 1/4 cup flour. Place dough on floured surface. With lightly floured hands, turn dough to coat with flour. Gently knead dough 8 to 10 times, working just enough flour into dough to prevent sticking. Divide dough in half. Flatten each half into 7-inch round. Cut each into 6 wedges; place on sprayed cookie sheet. Brush with 3 tablespoons milk.
Bake at 400°F. for 12 to 16 minutes or until light golden brown. Remove scones from cookie sheet; place on wire rack.
Place melted white chocolate in small resealable food storage plastic bag. Cut off small corner of bag. Pipe small amount of white chocolate over scones. Sprinkle with reserved 2 tablespoons apricots. Pipe with remaining white chocolate. Serve warm or cool.