In a large bowl, dissolve yeast in 1 cup warm water (95°–110°). Add 1 1/2 cups flour, sugar, and salt. Beat for 2 minutes with an electric mixer or wooden spoon. Add egg and butter and gradually beat in remaining 1 3/4 cups flour by hand until the batter is smooth. Cover with plastic wrap and refrigerate dough for at least 2 hours or up to overnight.
Turn dough out onto a well-floured board or counter. Roll dough 1/2 in. thick, flouring generously and turning dough 90° between rolls to keep it from sticking. Cut out rounds with a standard 2 1/2-in. doughnut cutter. Alternatively, cut rounds with a 2 1/2- or 3-in. round biscuit or cookie cutter, and cut out centers with a 1-in. round biscuit or cookie cutter. Place rounds and centers on 2 well-floured baking sheets at least 1 in. apart. Let dough rise in a warm place until slightly puffed, about 2 hours.
Put wire cooling racks over 2 empty baking sheets and set them near the stove. Pour oil into a large pot to a depth of 2 in. and heat to 325° to 350°. To get a feel for the method and how the dough should look and act, start by frying the holes. Working in batches of 6 to 8 holes and then 3 or 4 doughnuts, fry doughnuts until golden brown, turning once, about 1 minute each side. (Doughnuts should sink for 2 to 3 seconds before floating to the top; if they don't sink, the oil is too hot. Also, if they take much more or less than 1 minute per side to cook, adjust oil temperature.) As doughnuts brown, transfer with a slotted spoon to racks. After the first batch, test a doughnut hole by breaking it open. It should be light and cakelike inside, not greasy. If it's greasy, either the oil wasn't hot enough or it cooked too long. Be sure to fry the doughnuts only until golden brown.
While doughnuts are still slightly warm, dip in sugar or, as directed to 1963 readers, in âÂ€Âœyour favorite butter frosting (either chocolate or orange is especially suitable for Hallowe'en).âÂ€Â?
Note: Nutritional analysis is per doughnut.