Place cut potato in a medium saucepan and cover with cold water. Set saucepan over medium-high heat and bring to a boil, then reduce heat to a gentle simmer. Cook until potatoes are tender, about 20 minutes. Using a colander, drain potatoes. Set a ricer over now-empty pot and pass potatoes through; set aside. (See notes.)

Butter an 8- by 8-inch square aluminum baking pan and a large bowl.

For the Dough: In a medium bowl, whisk flour and yeast to combine; set aside.

In the bowl of a stand mixer, whisk sugar, water, potatoes, butter, egg, and salt to combine. Add flour and yeast mixture and, using a bench scraper, bring ingredients together and gently knead until a lumpy paste forms, about 10 seconds.

In a stand mixer fitted with dough hook attachment, mix dough on medium speed, scraping down sides of bowl as needed, until a sticky dough forms, about 5 minutes. (The dough will cling to the hook and sides of the bowl and resemble a soft, stretchy cake batter.) See notes below for instructions for making the dough by hand.

Using nonstick cooking spray, grease bench scraper. Use greased bench scraper to turn dough out into prepared bowl. Cover tightly with plastic wrap and allow dough to rise at room temperature until doubled in size, 1 ½ to 2 hours.

Using a greased bench scraper or greased hands, transfer dough to prepared baking pan. Gently press dough into prepared pan. Grease a piece of plastic wrap large enough to cover the baking pan; cover with plastic wrap, greased side down, and allow dough to rise until puffy and doubled in size, 45 minutes to 1 hour.

Adjust oven rack to middle position and preheat oven to 375°F (190°C).

For the Topping: In a small bowl, whisk brown sugar and cinnamon to combine. Dip fingertips in melted butter and make 25 to 30 deep indentations in the dough. Fill each indentation with a generous pinch of the sugar mixture, then press down gently with your index finger. Sprinkle dough with remaining sugar mixture. Drizzle thin rows of melted butter in a criss-cross pattern over dough, avoiding large pools or puddles as much as possible.

Bake in preheated oven until dough is a deep golden brown with crisp corners, 27 to 30 minutes. Remove cake from oven and let sit in pan on a wire rack until just cool enough to remove, about 20 minutes. Run a butter knife or an offset spatula around the edges of the pan and turn cake out onto wire rack and let cool until just slightly warm, about 30 minutes. Use a serrated knife to slice into squares. Serve warm.