Center a rack in the oven and preheat to 350°F.
Line a baking sheet with parchment or a silicone mat.
In a medium bowl, sift together the flour, cocoa, espresso powder, baking soda, baking powder, and salt. Set aside.
Working with a stand mixer, preferable fitted with a paddle attachment, or with a hand mixer in a large bowl, beat the butter and sugar together on medium speed until pale, about 2 minutes; the mixture may be crumbly.
Scrape down the sides of the bowl, add the eggs and vanilla and beat for another 2 minutes; don't worry if the mixture looks curdled.
Reduce the mixer speed to low and mix in the dry ingredients in 3 additions, mixing only until a dough forms. Scrape down the sides of the bowl.
Mix in the white and bittersweet chocolates.
Turn the dough onto a work surface. At this point the dough will be sticky. Spraying your hands with non stick cooking spray or rubbing them with a little bit of Canola oil will help make the dough more workable.
Knead in any dry ingredients that may have escaped mixing. Divide the dough in half. Place the dough on a parchment or silpat lined baking sheet and shape it into rectangles that measure 12-inches long, 2-3 inches wide and 1 inch tall.
Sprinkle each log with a little sugar, and bake about 25 minutes, or until the logs are just slightly firm. The logs will spread and crack - and that's just fine. Remove the baking sheet from the oven, put it on a cooling rack, and allow the logs to cool for about 20 minutes, leaving oven on.
Working with one log at a time, using a long serrated knife, cut each log into slices between 1/2- and 3/4-inch thick. Stand the slices up on the baking sheet and bake the cookies again, this time for just 10 minutes.
Transfer the biscotti to a rack to cool.