Begin by whisking together the large eggs and egg yolks in a standing mixer. Set the mixer to medium speed and whisk for about 5-6 minutes, or until the mixture becomes pale and thick, forming ribbons on the surface.
In a medium saucepan, combine the white sugar and water, stirring gently to ensure they're fully mixed. Heat over medium-high heat until the sugar dissolves completely, then heat until it reaches a temperature of 215-230°F.
Carefully pour the syrup into the egg yolk mixture while the mixer is running on low speed.
Increase the mixer to high speed and continue whisking until the bowl feels cool to the touch, around 10 minutes.
Gradually add the room-temperature unsalted butter, a chunk at a time, ensuring each piece is fully incorporated before adding the next.
If the buttercream curdles, chill the bowl or warm gently to rectify.
Finally, scrape the seeds from the vanilla beans into the buttercream and whisk until evenly distributed.