Beat 1 cup butter and sugar at medium speed with an electric mixer until light and fluffy.
Add 1/2 cup corn syrup and egg yolks, beating well. Gradually stir in flour; cover and chill 1 hour.
Melt 1/4 cup butter in a heavy saucepan over medium heat; stir in powdered sugar and 3 tablespoons corn syrup. Cook, stirring often, until mixture boils.
Remove from heat. Stir in pecans; chill 30 minutes. Shape mixture by 1/2 teaspoonfuls into 1/4-inch balls; set aside.
Shape cookie dough into 1-inch balls; place 2 inches apart on lightly greased baking sheets. Beat egg whites until foamy; brush on dough balls.
Bake at 375 for 6 minutes.
Remove from oven, and place pecan balls in center of each cookie.
Bake 8 to 10 more minutes or until lightly browned. Cool 5 minutes on baking pans; remove to wire racks to cool completely. Freeze up to 1 month, if desired.
