Preheat the oven to 350° and line 2 baking sheets with parchment paper. In a standing electric mixer fitted with the paddle, beat the butter with the sugar at medium speed until creamy. Add the eggs, vanilla seeds and rum and beat until smooth. In a small bowl, whisk the flour with the cocoa powder, baking powder, salt and cinnamon. Add the dry ingredients to the batter and beat at low speed until incorporated. Beat in the whole pistachios.
Divide the dough into 4 pieces. On a lightly floured surface, roll each piece into a 1 1/4-inch-thick log. Arrange 2 logs on each baking sheet about 3 inches apart and press to flatten slightly. Brush the logs with the egg white.
Bake the logs for about 25 minutes, until the tops are cracked and glossy and slightly firm to the touch. Let the logs cool slightly.
Using a sharp chef's knife, cut the logs into 1/3-inch-thick diagonal slices. Arrange the slices on the baking sheets; bake for about 20 minutes, until dry. Let cool completely. Partially dip the biscotti in the melted chocolate and sprinkle lightly with chopped pistachios. Refrigerate until the chocolate is set.