In the bowl of a freestanding electric mixer, combine 1 cup large egg whites and 2 1/4 cups granulated sugar. Mix with a heatproof rubber spatula to combine into a sugary slurry. Fill a medium saucepan with a few inches of water and place it on the stovetop over medium-high heat. Set the mixer bowl on top of the saucepan to create a kind of double boiler, making sure the mixer bowl doesn't touch the water.
Heat the egg and sugar mixture until it reaches 160 (F) on the candy thermometer, using the rubber spatula to scrape down the sides of the bowl to prevent any ingredients from burning and stirring the mixture occasionally. This process should take around 10 minutes, so if you find that it's taking longer, don't be afraid to crank up the heat!
Use oven mitts to carefully transfer the mixer bowl onto the stand mixer and fit a whisk attachment in place. Turn the mixer on to its highest setting and beat the mixture for 10 minutes, or until you've created a meringue with medium-stiff peaks and the sides of the bowl are only slightly warm or at room temperature.
Scrape down the whisk, and replace it with the paddle attachment. With the mixer on low speed, add 2 1/4 cups cubed unsalted butter a few cubes at a time, continuing to mix until the butter is incorporated. At this point, it should look like a thick soup, and that's totally okay. Add 2 teaspoons pure vanilla extract and a pinch of kosher salt, and turn the mixer up to its highest speed. Beat the mixture on high speed for about 2 to 3 minutes, or until a fluffy, silky, magical buttercream frosting has formed.