Make the dough at least 2 hours before baking the turnovers. Whisk together the two flours, baking powder and salt in a large bowl. Using an electric mixer at high speed, beat the butter and sugar until light in color and texture, about 3 minutes. One at a time, beat in two of the eggs (reserve the third for glazing), then the vanilla and ginger. Reduce the mixer speed to low. Add the flour mixture and mix until thoroughly combined. (If using a hand mixer, use a wooden spoon to stir in the flour.) Gather up the dough and divide in half. Flatten each portion into a thick square. Wrap each in plastic wrap and refrigerate until chilled and firm, at least 2 hours or overnight.
Position a rack in the center of the oven and preheat to 400 F. Line two baking sheets with parchment paper or a silicone baking mat. Whisk the remaining egg with 2 teaspoons of water.
Lightly flour a work surface. Working with one disk of dough at a time, roll out the dough into a 12-inch square. Cut into four 6-inch squares.
To make the filling, whisk together the brown sugar, flour, and lemon zest in a bowl. (To discourage the pineapple from giving off too many juices, chop it just before making the filling, and mix the pineapple with the flour mixture in two batches.) In another bowl, mix half of the flour mixture with half of both the pineapple and macadamia nuts.
Brush the edges of a square of dough with the egg mixture. Place a heaping 1/4 cup of the pineapple mixture in the center of the square. Fold the corners of the dough to overlap in the center of the filling, and press gently but firmly. Repeat with the remaining three squares of dough and filling. Brush the turnovers with the egg mixture and sprinkle with granulated sugar.
Place one sheet of turnovers in the oven. Bake until the turnovers are golden brown, 15 to 18 minutes. Transfer to a wire cake rack to cool. While the first tray is baking, roll out the second portion of dough, and fill with the second batch of the pineapple filling. Repeat the egg glaze and sugar topping with the second sheet.