Make the caramel: Pour 1/4 cup water into a medium heavy-bottomed saucepan. Add 1/2 cup plus 2 tablespoons sugar. Cook over medium-high heat, stirring occasionally, until sugar is dissolved and syrup is clear. Continue to cook, without stirring, until syrup comes to a boil, washing down sides of pan with wet pastry brush two or three times to prevent crystals from forming. Let syrup boil, gently swirling pan occasionally, until syrup is dark amber in color.
Immediately remove from heat; carefully pour in cream (caramel will bubble vigorously). Stir until smooth. Transfer to a bowl and let cool.
Whisk together remaining 1/2 cup plus 2 tablespoons sugar, egg whites, and salt in the heatproof bowl of a standing mixer. Set the bowl over pan of simmering water; whisk until sugar is dissolved and mixture registers 140 degrees on an instant-read thermometer.
Return the bowl to the mixer; fit mixer with whisk attachment. Beat on medium-high speed until stiff. Reduce speed to medium-low; add butter, a few tablespoons at a time. Continue whisking until smooth. Switch to the paddle attachment. Add caramel and vanilla; beat on lowest speed until combined, 3 to 5 minutes.