Whisk butter and egg yolks together in large heat-resistant bowl set over medium saucepan with 1/2 inch of barely simmering water (don't let bowl touch water) until mixture is smooth and homogeneous.
Slowly add 1/2 cup boiling water and cook, whisking constantly, until thickened and sauce registers 160 to 165 degrees on instant-read thermometer, 7 to 10 minutes. Off heat, stir in lemon juice and cayenne. Season with salt to taste.
Bring 6 cups of water to a boil in a large pot.
Add a tablespoon of vinegar and turn off the heat.
Poached Eggs
Add eggs and cook for 3 minutes. Transfer to another pot with water at 150 degrees to hold.