Heat the butter slowly in a pan. Keep it warm. You will see that the butter kind of separates, you get a clear part and some white stuff at the bottom. Leave the white stuff there, only use the clear butter.
Mix the egg yolks over very gentle heat (never go above 60° C / 140° F) with the mustard, vinegar, stock and tarragon in another pan. Whisk/mix until the mixture just begins to thicken. Remove it from the stove immediately. If the pan is thick, place it on a wet kitchen towel, so the towel will absorb any residual heat from the pan.
Now the hard part begins. Start by dripping the butter into the yolk mix, constantly whisking. After a while, pour the butter in a very thin stream into the yolks, constantly whisking. You want the egg yolks to bind/hold the butter in suspension.
Two things are of utter importance to the success of this sauce:1. Make sure the butter and yolk mix have the same temp before mixing. Use a thermometer! I started mixing when they were both at 40° C / 104° F.2. Don’t add too much butter at a time. You really should mix the butter very little at a time.
Now, what to do with all those egg whites, you wonder? Why not a Creme Toum, it’s a great sauce for grilled meats.