For the dipping sauce: Because the caramel takes a little while to cook, start by placing the sugar in a large, heavy skillet. Heat the sugar over medium heat, stirring with a fork until it begins to melt. Once it is melted, stop stirring. Cook the sauce, giving the skillet a swirl every now and again, so the sugar melts evenly. Cook until it is dark amber in color, and then remove from the heat. Keep an eye on the sugar because it can go from dark amber to burnt in a blink. 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, 2 tablespoons water, the bourbon, vinegar and salt and cook over low heat, stirring to incorporate all the ingredients and scraping up all the caramel from the bottom of the pan. Add the cream, bring to a boil and allow to cook for 1 to 2 minutes, stirring, and then remove from heat. Allow to cool slightly.
For the beignets: Set the oven rack in the middle of the oven and preheat the oven to 300 degrees F. Heat the oil in a medium heavy-bottomed pot over medium-high heat to 375 degrees F.
In a small bowl, beat the egg with a fork, stir in the sparkling cider and 1 tablespoon oil until nice and foamy. Cook's Note: Make sure you use a sparkling cider-it makes the batter lighter and fluffier.
In a large, shallow bowl, stir together the 1 1/4 cups flour and cinnamon if using. Make a well in the center of the flour (similar to the way you would if you were making pasta). Pour the cider mixture into the center of the well. Stir in a circular motion, with the fork slowly dragging in and incorporating the flour. Stir until a lumpy batter forms.
Put the remaining 1/2 cup flour in a plate or pan. Working in batches of 3, dredge the apple rings in the flour, shaking off the excess, and then dip in the batter to coat, letting the excess drip off. Carefully slip the rings into the hot oil 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 a paper towel-lined plate to drain the 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, then drizzle over the top and serve more on the side.