Bring potato and water to cover to boil in small saucepan. Reduce heat and simmer until potato is tender, about 15 minutes. Drain potato, then mash until smooth. Let cool completely, at least 30 minutes.
Transfer 1/2 cup mashed potato to medium bowl (discard remaining potato) and beat in egg, milk, 1/2 cup sugar, and vanilla until combined. Whisk flour, baking powder, salt, cinnamon, and nutmeg in large bowl. Make well in center of flour mixture and add potato mixture. Stir to form moist and sticky dough.
On heavily floured work surface, roll dough into 18 by 14-inch rectangle about 1/4 inch thick. Cut dough into strips, make slit in each strip, and twist to shape dough to resemble crossed legs (see photos). Transfer crullers to floured baking sheet and refrigerate until ready to fry. (Crullers may be covered with plastic wrap and refrigerated for up to 24 hours.)
Heat oil in large Dutch oven over medium heat until temperature reaches 350 degrees. Carefully lower 6 crullers into hot oil and fry, maintaining temperature between 325 and 350 degrees, until crisp and deep brown on both sides, 4 or 5 minutes. Using slotted spoon, transfer crullers to paper towel-lined plate and drain for 3 minutes. Toss crullers in bowl with remaining 3/4 cup sugar and transfer to serving plate. Repeat with remaining crullers, regulating oil temperature as necessary. Serve.