In a large bowl of an electric mixer, combine egg yolks, egg and salt. Beat
on highest speed until mixture is thick and drops softly from the beaters, 7
to 10 minutes,
Beat in sugar, a small amount at a time.
Beat in rum and vanilla extract.
Remove from mixer.
Fold flour into mixture by hand until incorporated.
Turn dough onto generously floured surface. Knead dough until blisters form
on the dough, about 10 minutes. Add small amounts of flour as needed to surface
to keep dough from sticking. Divide dough in half. Cover one half with inverted
bowl, towel or plastic wrap to prevent drying.
Roll out other half of dough as thin as possible, (about an 8 x 12-inch
rectangle). If dough resists, let it rest for a few minutes and resume rolling.
Cut dough into 2 x 4-inch rectangles. Make a 2-inch slit from center almost
to end of each dough strip. Pull opposite end of strip through slit to twist
the dough. Repeat with remaining dough.
In a large skillet, heat vegetable shortening until it reaches 350 degrees
F on a deep-fat fry thermometer. (This is important, if the oil is too cool,
dough will absorb too much oil; if it is too hot, the pastry will burn on the
outside and not cook properly on the inside.)
Add a few angel wings at a time and fry until golden on both sides, turning
about halfway through cooking time, about 1 1/2 minutes total.
Remove to drain on paper towels.
Repeat with remaining angel wings until all are fried.
Cool completely and dust liberally with confectioners' sugar.