Heat oil in a deep-fryer or large saucepan to 365 degrees F (185 degrees C). Mix 3 cups flour, 1 cup sugar, the baking powder, salt, and 1/2 packet of the soft drink mix together in a large bowl.
Whisk the milk, eggs, butter, and vanilla extract together in a separate large bowl until blended; slowly stir in the flour mixture until entirely incorporated in a batter the consistency of a thick pancake batter.
Drop the batter by large spoonfuls, about 2 teaspoons in size, into the preheated oil; fry until deep golden brown on all sides, 2 to 3 minutes. Remove to drain on a platter lined with paper towels.
Mix together 1 cup sugar with the remaining 1/2 packet of soft drink mix in a flat-bottomed dish. Roll the drained donuts in the sugar mixture while still hot. Set aside on a fresh set of paper towels to cool slightly; serve warm.