Make the doughnut cake: Place potatoes in a medium saucepan, cover with 2 cups cold water plus more as needed to cover potatoes; bring to a boil over high. Reduce temperature to medium-low and simmer, undisturbed, until potatoes are tender, 15 to 20 minutes. Reserve 7 tablespoons cooking water in a heatproof measuring cup, then drain potatoes and return to saucepan. Set potatoes aside.

Add cold milk to reserved 7 tablespoons cooking water in measuring cup and stir to combine. When it has cooled to lukewarm (100°F to 110°F), add yeast and whisk with a fork; set aside until foamy, about 5 minutes.

Mash potatoes in saucepan until smooth, then transfer ¾ cup mashed potatoes to the bowl of a stand mixer fitted with paddle attachment. Discard or reserve any remaining mashed potatoes for another use. Add sugar and 6 tablespoons butter and beat on low speed until butter has melted and mixture is smooth and creamy, about 30 seconds. Add egg yolks and salt; beat on medium-low speed until incorporated, about 10 seconds. Add yeast mixture and beat on medium-low speed until combined (The mixture may look a little curdled at this stage; don’t worry, it will come together again.)

Scrape down sides of bowl with a spatula and change to dough hook attachment. Add flour, about 1 cup at a time, mixing on low speed for a few seconds after each addition. When all the flour has been added, increase speed to medium and beat until dough mixture forms a rough ball, about 2 minutes

Scrape dough out onto a lightly floured work surface and knead gently by hand until dough is smooth, soft, and supple, about 2 minutes.

Grease a large bowl with 1 tablespoon butter and add dough. Cover with plastic wrap, and refrigerate until doubled in size, at least 8 hours or up to 24 hours. (If you are making the cake on the same day you want to eat it, place the covered bowl in a warm place for 1 to 2 hours, then proceed with the recipe once the dough has doubled in size.)

Once doubled in size, grease bottom and sides of a 9- x 13-inch baking pan with remaining 1 tablespoon butter. Punch down dough, and transfer to prepared baking pan. Gently press and stretch dough to fit bottom of pan, then cover loosely with a sheet of parchment paper. Let rise in a warm (75°F) place until doubled in size and puffy, about 2 hours. (If making cake on the same day you want to eat it, let rise in a warm place until doubled in size and puffy, about 30 minutes.)

While dough rises, preheat oven to 375˚F and make the topping: Stir together pecans or walnuts, brown sugar, cinnamon, and salt in a medium bowl. Set aside until ready to use.

Using wet fingers, press fingers into puffed-up dough to create deep dimples all over. Scatter the cold diced butter over dough, then sprinkle cinnamon-sugar mixture evenly over top. Drizzle cream all over, then bake in preheated oven until golden brown, puffed up, and bubbling, about 30 minutes. Carefully transfer to a wire rack; let cool in pan for about 30 minutes before cutting into large squares to serve.