In the heatproof bowl of a stand mixer, whisk together brown sugar, egg whites, and granulated sugar by hand. Place bowl over a saucepan of simmering water. Cook, whisking occasionally, until sugars completely dissolve and an instant-read thermometer registers 120°F (49°C) to 130°F (54°C).

Carefully return bowl to stand mixer. Using the whisk attachment, beat at high speed until bowl is cool to the touch, about 8 minutes. Add butter, 2 tablespoons (28 grams) at a time, beating until combined after each addition. Beat in vanilla and salt. Use immediately, or refrigerate in an airtight container for up to 3 days. If refrigerating, let stand at room temperature for 2 hours and rewhip until smooth before using.