Preheat oven to 350 degrees F (177 degrees C).
Line a baking sheet with parchment paper.
Using hand or stand mixer, beat the sugar and eggs on high speed until thick, pale, and fluffy (about 3-5 minutes). Add vanilla extract and beat for a few seconds more.
In a separate large bowl, whisk together the flour, baking powder and salt. Add the egg mixture and beat until combined.
Fold in the sour cherries, pistachios and cranberries.
Transfer the dough to a well floured counter and roll into a log shape, about 12 inches (30 cm) long and 3 1/2 inches (9 cm) wide. You may have to flour your hands to form the log as the dough is quite sticky. Alternatively, you can wet your hands to for the log without adding extra flour.
Whisk one egg with a splash of cold water or milk until pale yellow and perfectly mixed.
Using a pastry brush paint the log with a light, even coat of egg wash.
Place the log on the baking sheet and bake on the middle rack for about 30 minutes or until firm to the touch.
Remove the log from the oven and let cool on a wire rack for about 10 minutes.
Reduce oven temperature to 325 degrees F (165 degrees C).
Transfer the log to a cutting board and cut into about 1/2 inch (1.25 cm) slices.
Place the biscotti, cut side down, on the baking sheet. Bake for about 8-10 minutes, turn slices over, and bake for another 8-10 minutes or until golden brown.
Remove from oven and let cool.
Can be stored in an airtight container for several weeks.
Makes about 20 - 24 biscotti.