1. Place the eggs, apple cider, milk, melted butter, and vanilla in large bowl and whisk to combine. Whisk in the sugar, and set aside.
2. Measure 1 1/2 cups of the self-rising flour into a small bowl. Stir in the cinnamon. Toss the apples with the remaining 1/2 cup flour. Place the flour mixture in the bowl with the egg mixture and whisk until it is combined and smooth, about 1 minute. Fold in the floured apples until combined.
3. Place enough peanut oil in a large, deep cast iron skillet or Dutch oven to measure 2 inches. Place the pan over medium-high heat, and bring to 350 to 365 degrees F. When the oil is hot, drop generous tablespoons of the batter into the pan, frying about four doughnuts at a time. Let the doughnuts fry about 1 1/2 to 2 minutes per side, turn, and continue cooking 1 1/2 to 2 minutes on the other side. Remove to a wire rack set on top of brown paper to drain.
4. Repeat with the remaining batter, giving the oil time to reheat to 350 to 365 degrees before frying. With a slotted spoon or sieve, clean up the oil between frying, removing any burned bits from the oil. Dust the warm, drained doughnuts with confectioners’ sugar and serve.
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