Combine 1/2 cup water, the butter and vanilla in a medium saucepan; bring to a simmer, stirring to melt the butter. Remove from the heat and stir in the flour, instant potatoes, 1/2 cup sugar and the salt. Return to medium heat and stir until the dough is tight and doesn't stick to the sides of the pan, about 2 minutes. Transfer the dough to a stand mixer or a medium bowl; cool for about 10 minutes.
Beat the dough with the paddle attachment on medium-high or with a wooden spoon; add the eggs and egg white one at a time, beating until each egg is fully incorporated before adding the next. Once the eggs are mixed in, beat until smooth and glossy, about 1 minute. Set aside for 1 hour.
Combine the remaining 1 cup sugar and the cinnamon in a shallow dish. Place a tall heavy-bottomed pot over medium heat and fill it about one-third of the way with oil; heat until a deep-fry thermometer registers 340 degrees. With a small ice cream scoop, drop heaping teaspoonfuls of dough into the oil (in batches). Fry, turning once, until golden brown, about 6 minutes. Transfer the doughnuts with a slotted spoon to the cinnamon-sugar mixture and roll to coat. Serve warm.
Photograph by Tina Rupp