1.

Place potatoes and 3 1/4 cups water in a large saucepan. Bring to a boil. Cook until potatoes are fork tender, about 12 minutes. Remove from heat; drain, reserving liquid. Return potatoes to saucepan. Using a whisk, mash potatoes over medium-high heat until dry, about 1 minute. Set 1 cup potatoes aside; reserve remainder for another use.

2.

Place yeast in bowl of an electric mixer; pour 3/4 cup reserved potato liquid over yeast. Sprinkle granulated sugar over yeast; let dissolve, 2 to 3 minutes. Add 1/2 cup all-purpose flour and 1/2 cup cake flour; stir with a wooden spoon until combined. Cover with plastic wrap; set aside in a warm place until surface is covered with bubbles, about 30 minutes.

3.

Place chocolate in a heat-proof bowl or the top of a double boiler, and set over a pan of simmering water. Heat, stirring with a rubber spatula, until melted and smooth.

4.

Place 1/2 cup plus 1 tablespoon reserved potato liquid in a small saucepan; bring to a boil over medium-high heat. Remove from heat, and stir in the baking soda and cocoa powder until combined. Add the melted chocolate, the reserved cup mashed potato, confectioners sugar, and salt; stir to combine.

5.

Add potato mixture, 1/2 cup all-purpose flour, and remaining 1/2 cup cake flour to yeast mixture. Using paddle attachment, mix on low speed until flour has been moistened, about 30 seconds. Increase speed to medium high; mix until thoroughly combined, 1 minute. Add the remaining 1/2 cup all-purpose flour; mix on low speed 30 seconds. Increase speed to high; mix another 30 seconds; the dough will be very sticky.

6.

Lightly oil a large bowl; set aside. Sift an even layer of all-purpose flour over a clean work surface. Turn dough out onto floured surface, and gather into a ball. Transfer dough to oiled bowl, and cover with plastic wrap. Set aside in a warm place to rise until doubled in size, about 1 1/2 hours.

7.

Turn out dough onto floured surface, and fold in half, deflating dough. Return dough to oiled bowl, re-cover, and set aside to rise until doubled in size again, about 30 minutes.

8.

Heat oil in a low-sided six-quart saucepan over medium-high heat until a deep-frying thermometer registers 375 degrees. Lightly dust a baking pan with flour, and line a second one with paper towels; set both aside.

9. Sift a light coat of flour onto a clean work surface. Turn out dough. Sift another layer of flour over dough. Roll into a 12-by-14-inch rectangle, about 1/2 inch thick. Using a 2 3/4-inch doughnut cutter, cut out doughnuts as close together as possible, dipping the cutter in flour before each cut. Transfer the doughnuts to the floured baking pan, and set aside 10 minutes, but not more.

10. Gently pull on sides of the doughnuts, slightly enlarging holes. Working in batches of four, transfer doughnuts to oil, and cook until tops have expanded and cracked slightly, about 2 minutes. Using a slotted spoon, turn doughnuts over, and cook 1 minute more. Transfer doughnuts to lined baking pan, and set aside until cool enough to handle. Repeat with remaining doughnuts.

11. Gather remaining dough scraps into a ball. Let rest 10 minutes; pat into a 1/2-inch-thick rectangle. Cut, let rest 10 minutes, and cook.

12. While still warm, coat doughnuts with chocolate glaze, and sprinkle with chopped pistachios or coconut, if desired. Transfer to a wire rack to set, and serve.