CRUST: Lightly butter an 8-inch square baking pan. Line bottom and 2 sides of pan with parchment, leaving a 1-inch overhang; lightly butter the parchment.
Combine the flour, brown sugar, and salt in a food processor, and pulse a few times to combine. Add butter and pulse until largest pieces are the size of peas. Add coconut and white chocolate and pulse a few more times, just until mixture holds together. Transfer mixture to pan, and pack down firmly with the bottom of a measuring cup. Bake in a preheated 400-degree oven until crust is deep golden brown, 18 to 20 minutes. Remove from oven and reduce heat to 300 degrees.
FILLING: Whisk together eggs and granulated sugar in a bowl, then whisk in flour. Gently stir in lime juice (do not whisk or stir vigorously or batter will become foamy).
Pour filling over hot crust, and bake until filling is set, about 15 minutes. Let cool completely on a wire rack.
Run the tip of a knife along the 2 sides of the pan NOT lined with parchment overhang. Transfer to a cutting board, dust with confectioners' sugar, and cut into 1-inch squares. Bars can be stored in an airtight container at room temperature for 2 days.