Chop 1 cup of butter and place in a medium saucepan. Melt the butter over medium heat. Turn the heat to low and add the chai tea. Continue to simmer the butter mixture for 5 to 10 minutes. Remove from the heat and allow the butter to cool and tea to steep for another 10 minutes.
Strain the infused butter with a mesh sieve. Press down on the tea to extract as much of the butter as possible. It is okay if a little bit of tea remains in the butter. Chill the butter in the refrigerator until it is the consistency of softened butter.
Meanwhile, allow the remaining butter to come to room temperature.
Place the egg whites and sugar in the bowl of a stand mixer and whisk together (by hand) until combined.
Fill a saucepan with a few inches of water and bring to a boil. Reduce the heat and place the bowl with the egg white mixture on top to create a double-boiler. Make sure that the water remains at a simmer and does not touch the bottom of the bowl.
Stirring intermittently, heat the egg mixture until hot to the touch or 160°F on a candy thermometer.
Carefully remove the bowl and place it on the stand mixer fitted with a whisk attachment. Whisk on high speed until stiff peaks form and the outside of the mixing bowl returns to room temperature, about 8 minutes.
With the mixer on low, add in the softened butter, a couple tablespoons at time, followed by the chai-infused butter. Once all of the butter has been added, stop the mixer and switch the whisk for the paddle attachment.
Mix the buttercream on medium speed until the butter is fully incorporated. Increase the speed to medium-high and mix until fluffy, about 5 minutes. Add the vanilla bean paste and mix to combine. Decrease the speed to low and mix until silky and smooth, about 2 minutes.