Preheat oven to 325°F. Spread coconut on rimmed baking sheet. Bake until coconut is light golden, stirring occasionally, about 8 minutes. Cool completely.

Using electric mixer, beat butter and sugars in large bowl until well blended. Mix in salt and vanilla. Beat in flour in 2 additions. Stir in toasted coconut. Gather dough together; divide in half. Place each half into a one-gallon Ziplock bag. Flatten each half into an even thickness about ¼-inch, spreading the dough to the very edges and corners of the bags. Chill flat, at least 1 hour. DO AHEAD Can be prepared 2 days ahead. Keep chilled.

Preheat oven to 325°F. Line 2 rimmed baking sheets with parchment paper. Carefully split open the Ziplock bags along the side seams. Using 1¾- to 2-inch (approximate size) cookie cutter, cut the dough into shapes. Transfer cookies to prepared baking sheets, spacing 1 inch apart. Gather dough scraps and reroll; cut out additional cookies. Repeat procedure with remaining dough disk until all of dough is used. Place pans in the refrigerator for 5-10 minutes to cool and firm.

Remove the pans from the refrigerator and lightly sprinkle coconut sugar over the tops of the cookies. Bake cookies until light golden, about 18 minutes, rotating the pans halfway through baking. Cool on baking sheets 10 minutes. Transfer cookies to racks and cool completely.