Put the egg whites into the bowl of a stand mixer with the whisk attachment. I would highly suggest using a stand mixer to make this recipe. It takes a long time to mix the meringue so your arm will get very tired if using a hand mixer.
Place the larger amount of granulated sugar (i.e., 300g) into a small saucepan, add the water, and stir with a spatula to moisten the sugar. Continue stirring until the sugar has dissolved. Place the candy thermometer in the saucepan and do not stir the sugar mixture anymore. Stirring at this point can lead to crystallization. Bring the mixture to a simmer over medium-high heat until it reaches 230°F.
While letting the sugar continue to cook on the stove, turn the stand mixer to medium and gradually pour in the remaining granulated sugar. Whip the egg whites until they form loose peaks and then turn the mixer to low speed until the sugar is ready.
When the sugar reaches 248°F/120C, remove the pan from the heat and set the thermometer on a heat-safe surface. Turn the stand mixer to medium-low speed and slowly pour the sugar between the side of the bowl and the whisk. Try very hard to avoid pouring the sugar onto the whisk because it will whip the sugar onto the side of the bowl and create hard strings of cooled sugar.
After the sugar has been poured into the mixing bowl, increase the speed to medium-high and whisk for 15 minutes, or until the bottom of the bowl is at room temperature and the whites hold stiff peaks. Warm egg whites will melt the butter and cause it to become the wrong texture (i.e., a soupy mess).
Reduce the speed to medium and add the butter, a few pieces at a time. If it looks like the mixture is broken (i.e., if the buttercream looks curdled), increase the speed of the mixer until it returns to the correct consistency. If the consistency of the buttercream is too loose after adding all the butter, place the buttercream in the fridge for 10-15 minutes and then whip it again.
Add the vanilla extract and mix to combine.
This recipe makes 450g/3 cups of buttercream.
I generally multiple this recipe by 3 if I am covering a two layered 9-inch cake. I sometimes multiply it by 4 to make sure I have enough buttercream for special decorations (e.g., piped swirls or borders like on my Confetti Cake).