Combine flour, baking soda, baking powder, and salt; stir well until blended.
Beat 1 cup shortening at medium speed with an electric mixer until creamy; gradually add sugars, beating well. Add eggs and vanilla; beat well. Add flour mixture, mixing just until blended. Stir in oats and cornflakes.
Drop dough by heaping tablespoonfuls, 2 apart, onto lightly greased baking sheets; flatten slightly. Bake at 325° for 12 to 14 minutes. Cool slightly on baking sheets. Transfer cookies to wire racks to cool completely.
Microwave white chocolate and 3 Tbsp. shortening in a medium-size microwave-safe bowl at HIGH 1 minute or until white chocolate melts, stirring once. Stir in peppermint extract. Dip flat bottom of each cookie into melted white chocolate, letting excess drip back into bowl. Place dipped cookies, dipped side up, on wax paper; let stand 1 hour or until white chocolate sets.