In a medium saucepan, combine sugar and ½ cup (120 grams) water. Bring to a boil over high heat, being careful not to splash sides of pan and stirring to help sugar dissolve. (Do not stir once mixture starts to boil.) Cook until an instant-read thermometer registers 240°F (116°C).

Meanwhile, in the bowl of a stand mixer fitted with the whisk attachment, beat egg whites, cream of tartar, and salt at medium speed until soft peaks form.

With mixer on medium speed, pour hot sugar syrup between side of bowl and whisk attachment in a slow, steady stream. Increase mixer speed to high; beat until stiff peaks form and bowl and meringue are relatively cool to the touch, 8 to 10 minutes.

Switch to the paddle attachment. With mixer on medium speed, gradually add butter, a few cubes at a time, beating until combined after each addition. (Mixture will look very loose to start, and may even look broken at one point, but buttercream will come together. [See Notes.]) Beat in vanilla bean paste. Beat until light and fluffy, 1 to 2 minutes, stopping to scrape sides of bowl. Use immediately, or refrigerate in an airtight container. Let refrigerated buttercream come to room temperature, and rewhip using the whisk attachment until silky smooth before using. (See Notes.)