In a mixer fitted with the paddle, mix eggs, granulated sugar, salt, and vanilla on low until combined, about 1 minute.
Put flour and butter in a medium bowl. Using a pastry blender or your fingers, blend until mixture resembles coarse meal. Make a well in center of flour mixture; pour in egg mixture. Gently combine with a rubber spatula until a sticky dough forms.
Transfer dough to a lightly floured surface. Knead until smooth and no longer sticky, 4 to 5 minutes; if necessary, add more flour. Cover with plastic wrap, and let rest 30 minutes.
Divide dough into 4 sections. Put one on floured work surface; cover remainder with plastic. Roll dough out to a very thin 7-by-16-inch rectangle. Using a fluted pastry wheel, trim to a 6-by-15-inch rectangle; cut into fifteen 1-by-6-inch strips. Tie each strip into a knot. Repeat with remaining dough.
Put oil in a small, deep saucepan. Heat over medium heat until a deep-fry thermometer registers 350 degrees. Line a baking sheet with paper towels. Using a slotted spoon, transfer 5 knots at a time to the hot oil. Fry until golden, about 2 minutes, turning once. Transfer to baking sheet to drain. Sift confectioners' sugar over cookies.