Place the egg whites in the bowl of a stand mixer fitted with the whisk attachment. Do not turn mixer on yet.
Place the 150 grams of sugar in a small saucepan and add the water, stirring until moistened. Bring to a simmer over medium-high heat, stirring occasionally and simmer until the syrup reaches 230 degrees F.
Letting the syrup continue to cook, turn the mixer to medium speed, gradually pour the remaining 33 grams of sugar into the whites, and whip until the whites are beginning to form very loose peaks. If the whites are ready before the syrup reaches 248 degrees F, turn the mixer to the lowest setting just to keep the moving.
When the syrup reaches 248 degrees F, remove the pan from the heat. Turn the mixer to medium-low and slowly add the syrup to the whites, pouring in between the side of the bowl and the whisk. 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 a stiff peak.
Reduce the speed to medium and add the butter, a few pieces at a time. Add vanilla. If at any time the mixture looks "broken" or curdled, increase the speed and beat to re-emulsify it, then reduce the speed and continue adding the butter. If the butter is too loose to hold its shape, it can be refrigerated for up to a few hours to harden, then beaten again to return to its proper consistency.
** The buttercream can be stored in a covered container in the fridge for up to 5 days or frozen up to 1 month. If you make this ahead of time, bring the buttercream to room temperature in the bowl of a stand mixer. Then beat on low to return the buttercream to the proper consistency for piping or spreading.