In a medium bowl, whisk the flour with the baking soda, baking powder and salt. In a large bowl, using an electric mixer, beat the butter with the brown sugar at medium speed until blended. Beat in the granulated sugar until light and fluffy, about 2 minutes. Beat in the whole egg, egg white and vanilla. At low speed, beat in the dry ingredients, in 2 additions. With a large wooden spoon, mix in the oats, semisweet and white-chocolate chips and the coconut. Cover the cookie dough and refrigerate for 1 hour, until chilled.

Preheat the oven to 375°. Line 3 large baking sheets with parchment paper. Using a rounded tablespoon, spoon mounds of dough onto the baking sheets, at least 3 inches apart. Bake the cookies for about 14 minutes, until golden and just set. Let the cookies cool on the baking sheets for 5 minutes, then transfer them to a rack to cool completely.