In a large bowl, beat the eggs and butter with 2 1/2 cups of the sugar until moistened. Beat in the milk and salt.
In a large saucepan, cook the remaining 1/2 cup sugar over moderately high heat, without stirring, until an amber caramel forms, about 7 minutes. Gradually pour the caramel into the egg mixture, stirring constantly; bits of the frosting will harden but will melt again later.
Scrape the frosting into the saucepan and cook over moderate heat, stirring frequently, until the temperature reaches 230° on a candy thermometer, about 20 minutes. The frosting should be thick and dark amber.
Transfer the frosting to a large heatproof bowl. Using an electric mixer, beat the frosting at medium-high speed until thickened, pale and smooth, but still warm and spreadable, about 8 minutes; the frosting should resemble creamy peanut butter. Use the frosting right away; if it hardens, rewarm it over low heat, stirring, just until spreadable.