Heat sugar in a 12-inch heavy skillet over medium heat, stirring with a fork to heat sugar evenly, until it starts to melt, then stop stirring and cook, swirling skillet occasionally so sugar melts evenly, until it is dark amber. Stir in butter, water, rum, vinegar, and a pinch of salt (caramel will harden and steam vigorously) and cook over low heat, stirring occasionally, until caramel has dissolved.
Add cream and bring to a boil, stirring, then remove from heat. Cool to warm.
Preheat oven to 250°F with rack in middle. Set a cooling rack in a large shallow baking pan.
Heat 2 inches oil in a 5- to 6-quart heavy pot over medium-high heat to 375°F.
Meanwhile, peel apples and cut crosswise into 1/4-inch-thick slices.
Cut out core with cutter, then pat apple rings dry.
Put 1 1/4 cups flour in a large shallow bowl and make a well in center. Beat egg in a small bowl with a fork, then stir in cider and 1 tablespoon oil and pour into well. Stir with fork until a lumpy batter forms.
Working in batches of 3 or 4, dredge apple rings in remaining 1/2 cup flour, shaking off excess, then dip in batter to coat, letting 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.
Transfer to rack and keep warm in oven. Return oil to 375°F between batches.
Just before serving, dust warm beignets with confectioners sugar. Stir sauce, then serve on the side.
• If you can't find self-rising cake flour, you can substitute self-rising all-purpose. Use 1 cup in the batter and 1/2 cup for dredging. Increase cider to 1 cup and follow recipe as directed.• Sauce can be made 3 days ahead and chilled, covered. Warm before serving.• Beignets are best freshly made but can be fried 2 hours ahead and kept, loosely covered, at room temperature. Reheat beignets (they should not touch) on a rack set in a large shallow baking pan, uncovered, in a 325°F oven until hot, 15 to 20 minutes.
