Place 2 cups shredded coconut in a food processor with sugar, salt and baking powder. Process until finely ground. Add butter and process just until no lumps remain. Add 1 whole egg and 1 egg yolk and vanilla. Process just until smooth. Add flour and pulse until a crumbly dough begins to form, being careful not to overmix.
Lightly whip the remaining egg white and place in a shallow bowl or on a plate. Place remaining 2 cups shredded coconut in another similar container. Scoop dough using a tablespoon and shape into balls. Place in egg white then roll in shredded coconut to coat.
Place cookies 2 inches apart on ungreased baking sheets. Bake until starting to brown but still soft to the touch, about 12-15 minutes. Transfer to wire rack to cool. Store at room temperature in an airtight container for up to 3 days.