Place the oven rack in the middle of the oven and preheat the oven to 250 degrees F.
Peel and core the apples and cut crosswise into 1/4-inch-thick slices. Pat apple rings dry.
In a large, shallow bowl, put 1 1/4 cups of flour and make a well in the center. In a small bowl, beat the egg with a fork, stir in the cider and 1 tablespoon oil, and pour into the well. Stir in a circular motion with the fork slowly incorporating flour, and stir until a lumpy batter forms.
Over medium-high heat, heat 2 inches of oil in a 5 to 6-quart heavy pot to 375 degrees F. Line a cookie plate with paper towels.
Working in batches of 3, dredge the apple rings in the remaining 1/2 cup flour, shaking off excess, and then dip in the batter to coat, letting the excess drip off, and fry, gently turning over once with a slotted spoon, until golden brown on both sides, about 2 minutes total per batch. Lay the beignets on the plate and allow the paper towel to absorb excess oil. Transfer to a cookie sheet and keep warm in the oven. Return the oil to 375 degrees F between batches.
Just before serving, dust the warm beignets with confectioners' sugar. Make sure the Butter Bourbon Dipping Sauce is warm, and serve on the side.
In a large, heavy skillet, heat the sugar over medium heat, stirring with a fork until it begins to melt. Once it is melting, stop stirring and cook, swirling the skillet occasionally so the sugar melts evenly. Cook until it is dark amber in color, and then remove from the heat. Put an oven mitt on the hand you will be using to stir in the ingredients and stand back to avoid splattering. Using a wooden spoon, stir in the butter, water, bourbon, vinegar, and salt and cook over low heat, stirring occasionally, until caramel has dissolved. Add the cream and bring to a boil and allow to cook for 1 to 2 minutes, stirring, and then remove from heat. Allow to cool down.;