Position 2 racks in the middle of the oven; preheat the oven to 325. Generously grease 2 baking sheets with butter (about 2 tablespoons) and dust with flour; tap off the excess flour.
Measure out the flour and put in a fine-mesh strainer set in a bowl. Beat the egg yolks and 1/2 cup sugar in a stand mixer fitted with the paddle attachment on medium-high speed until thick and pale, 6 to 7 minutes. Add the vanilla and beat until just combined. Transfer the egg yolk mixture to a large bowl. Clean out the mixer and dry.
Put the egg whites and salt in the mixer; beat with the whisk attachment on medium speed, adding the remaining 1/2 cup plus 2 tablespoons sugar, 1 tablespoonful at a time, until combined, about 3 minutes. Increase the speed to medium high and continue beating until the egg whites are shiny and stiff, 4 to 6 minutes.
Add one-third of the egg white mixture to the yolk mixture, then sift about one-third of the flour over the eggs. Gently fold the ingredients together with a rubber spatula. Repeat 2 more times, folding in one-third each of the egg white mixture and flour each time. (Do not overmix or the batter will deflate.) Put the confectioners' sugar in the mesh strainer.
Transfer the batter to a pastry bag fitted with a 1/2-inch round tip. Pipe 3-inch ladyfingers onto the prepared baking sheets, about 11/2 inches apart. Dust the ladyfingers with some of the confectioners' sugar to coat and set aside until the sugar starts to dissolve, about 5 minutes. Dust with another coating of confectioners' sugar. Bake until golden, 20 to 25 minutes, rotating the baking sheets halfway through the cooking time. Remove the ladyfingers with a spatula and transfer to a rack to cool.