In a saucepan (though I've used a skillet and even a crepe pan), boil togehter:

1/4 c vinegar (I use wine vinegar; type doesn’t matter, in my opinion)
1/4 c white wine
1 T minced green onion (use white part first - - or get fancy and use a shallot)
1 T tarragon (I use dried. If you use fresh, start with 1/2 T)

Boil until about 2 t remain. (More than 1 T [3 t] will result in a thin sauce.) Don't let the solids burn! Pour through fine sieve, throwing out the solids. I usually set the sieve directly over the blender jar (with blade assembly in place, of course). Press solids with a spoon to squeeze out all of the infusion.

When cooled (about 10 minutes - - though the infusion can wait as long as needed), add and whir:

3 egg yolks

Melt 2 sticks butter (1 c melted). I use 1 stick and top up with olive oil. I've never used straight olive oil, but why not? While melting, be careful not to splatter it all over the microwave! I melt it in 10-second pulses ahead of time and "hold" it on power setting 1. It's nice and warm and creates a warm sauce.

Don't make sauce with melted butter that's cooled down! Must be very warm or hot.

Dribble in hot melted butter/oil. This is the tricky bit: add the hot butter in droplets until you have about 3 T added. Then you can progress to a thin stream. Continue in this way until you hear the blender "change speeds." This means the emulsion of the egg yolks and butter has been made, and you can pour in the rest of the butter slowly (as opposed to upending the measuring cup).

Yum!

Notes:

• Allow 3 minutes from the time you start the sauce to serving time. If you're steaming broccoli (the traditional Béarnaise Transporter), get it going and close to done before the 3-minute mark.
• I use salted butter and don't add extra salt.
• Supposedly, one is not to use the "white solids" at the bottom of the melted butter, but I always do. What? I should throw this out?!
• Put the lid on the blender and remove only the little cap in the middle of the lid to minimize splattering. I "tent" a dish towel around the opening and pour the butter down through the "collar." This helps a little, but if you're in fancy clothing, you probably want to put on an apron for this step. (Or a chef's jacket, if, like me, you are especially piggy when you cook!)
• I have used dried minced onion in place of the green onion with no problem. Taste isn't greatly altered.
• Don't start with too much infusion. This puts you at a disadvantage for thickening! No more than 1 T! Boil it down!
• Sometimes I eliminate the white wine in the infusion and substitute 1 T more vinegar. I don't see that it makes that much difference. Since there's less liquid, I have to watch the infusion more carefully so that it doesn't burn. You also can skip the vinegar and use only white wine, but this removes the acidic "bite" from the sauce, in my opinion. You might like it better this way! I recommend you do it with Julia's proportions the first time.
• I don't know if this works in the food processor. Sorry.

Troubleshooting:

• Sauce is too thick: Thin with a little oil or melted butter (do not use water!).
• Sauce won't thicken: Your problem is that you added the melted butter too quickly. (Ahem. I told you to dribble it in....) Pour the sauce out of the blender jar except for 1 T. Add 1 T lemon juice. Start the blender. Add the sauce just as if you were adding melted butter.
• Sauce curdles ("turns"): Use same technique as if the sauce won't thicken.