Preheat oven to 350 degrees. In a large bowl, whisk together flour, 3/4 cup sugar, baking powder, and salt; stir in pistachios and chocolate. In a small bowl, whisk together eggs, butter, vanilla, and cocoa. Add egg mixture to flour mixture and stir until combined (dough will be stiff, so mix with hands if necessary).
Line a rimmed baking sheet with parchment and sprinkle with 2 tablespoons sugar. Divide dough in half and place on sheet. Shape into two 2 1/2-by-12-inch logs and sprinkle top of each with 1 tablespoon sugar. Bake until risen and firm, 15 to 20 minutes. Let logs cool completely on sheet, about 30 minutes.
Reduce oven to 300 degrees. Using a serrated knife, cut logs crosswise into 1/2-inch-thick slices. Arrange slices in a single layer on baking sheets. Bake until biscotti are dry, 15 to 20 minutes. Transfer to a wire rack to cool completely.
Keep in an airtight container at room temperature, up to 1 week.