Add dough ingredients to machine bread pan in order as per 1. manufacturer's instructions.
Set to"dough" mode.
Pour equal parts of vegetable oil and olive oil into a large frying pan to reach a depth of 2 inches. Heat the oil over medium heat until a deep-fry thermometer registers 375 degrees F.
Roll out the dough on a lightly floured surface to 1/2-inch thickness. Using a floured 2-inch cookie cutter, cut out doughnut rounds. Using a floured 1-inch cookie cutter, cut out a hole in the center of each doughnut. Gather the dough scraps and re-roll to cut out more doughnuts.
Whisk the sugar and cinnamon in a medium bowl to blend. Set the cinnamon-sugar aside.
When the oil is hot, working in batches, fry the doughnuts until they puff but are still pale, about 45 seconds per side. Using a slotted spoon, transfer the doughnuts to paper towels to drain. Cool slightly. While the doughnuts are still warm, add to the bowl with cinnamon-sugar and generously coat with the cinnamon-sugar. Serve warm.