In deep fryer or 2-quart heavy saucepan, heat oil over medium heat to 350°F.
Separate crescent dough into 4 rectangles. Firmly press perforations to seal. Stack 2 rectangles on top of one another. Fold in half widthwise to make tall stack. Repeat with remaining 2 rectangles.
To make 2 doughnuts, use 3-inch biscuit cutter to cut 1 round from each stack; use 1/2-inch biscuit cutter to cut small hole in center of each round. Reroll remaining dough to cut third doughnut.
Fry doughnuts in hot oil 1 1/2 minutes on each side or until deep golden brown and cooked through. Drain on paper towels. Cool 5 minutes.
Carefully split doughnuts in half. Place pudding in decorating bag fitted with tip, and pipe some of the pudding onto bottom half of each doughnut. Top each with some of the caramel sauce; sprinkle with salt. Cover each with top of doughnut.
In small bowl, mix powdered sugar and enough milk for spreading consistency. Spread on tops of doughnuts. Drizzle with additional caramel sauce.