In the bowl of a stand mixer fitted with the whisk attachment, add the egg yolks. Beat for 6 minutes on medium-high speed. The mixture will look thick and have a pale yellow appearance.
While the yolks are being whipped, combine the sugar and the 3 tablespoons of water in a medium saucepan and place over medium-high heat. Swirl occasionally until the sugar is dissolved.
Add 4 Tbsp of Kahlua.
Insert candy thermometer, and bring to a boil. Continue boiling the sugar-Kahlua mixture until it reaches 240F. Immediately remove from the heat.
With the mixer running, carefully and slowly pour the sugar syrup into the egg yolks in a steady stream. Small beads of sugar crystals may splatter onto the side of the bowl and harden. That's OK. Just ignore them.
Beat mixture on medium-high speed until very stiff, about 10 minutes.
Decrease the speed to medium and add the room-temperature butter, a tablespoon at a time, waiting a few seconds between each addition.
Add 2 more Tablespoons of Kahlua.
Beat on medium-high speed for about 3-5 minutes until thick and pipeable.