The béarnaise sauce is simple enough to make: stir the vinegar, champagne or wine, minced shallot, dried tarragon (and tarragon stems, if you’re using them), and salt in small saucepan. Place over medium heat and bring to a boil. Reduce the heat to low and simmer until the liquid reduces down to about 1 tablespoon—roughly 3 - 5 minutes.
Strain the liquid through a fine-mesh sieve, and into either a blender or short, but wide mason jar (that an immersion blender will fit into). Slide the egg yolks into the blender and blend on high speed for about 10 seconds (or stick the immersion blender in the jar and let it rip for 10 seconds, plunging it up and down a bit as you go).
Cut the butter up into cubes and melt in a small saucepan over medium heat until it's bubbly not, but not browning in the least.
Blend the yolks on high for a few more seconds and then, with the blender still running, stream the butter in (plunging the immersion blender up and down a bit). Once it's smooth, thick, and perfectly emulsified, stir in the fresh minced tarragon and lemon juice. Check it for seasoning and add more salt as need.
When it comes to the steak, I think something a little thicker is needed—1/2" minimum. What's important here isn't necessarily the cut of beef itself but, rather, the thickness and whether or not it has a bone (see recipe note for more on that).
When I make this, what I typically do is salt the beef and let it come to room temperature for about an hour on a wire rack set over a sheet tray. I put a cast iron pan in the oven, set it to 450° and leave it in there for 30 minutes once it comes to temp. I start the béarnaise, stopping just before melting the butter. Smear the steak in oil, heat the cast iron on the stove over medium-high for about 5 minutes and then slap the meat in the pan to sear it. This is when I start to melt the butter.
If the steak is 1" thick, I flip it over and immediately top it with a pad of butter and slide into the oven; if it's less than 1" I usually just cook the whole thing on the stove (see recipe notes below). Either way, during all this is when I emulsify the butter into the egg yolks, and stir in the tarragon and lemon. By the time the steak is finished resting, the sauce is done (and, if you throw a russet potato in the oven when the pan goes in and you keep everything in there a bit longer, you'll have a good starchy side, too),
The béarnaise should be fairly warm when served, but you can make it a bit in advance—I would say no more than an hour. Just leave it, covered, on the counter (away from the stove so the heat doesn't cause it to separate). When you're ready, splash in boiling hot water (2 - 3 t-spoons max) and blend again.
The steak pictured here is a strip steak (aka New York strip, Kansas City strip, or sirloin strip steak). They are sometimes marked as "Boneless T-Bone" or "Boneless Porterhouse", the strip being on one side of the bone, tenderloin on the other. The difference between a real T-Bone and Porterhouse is that the latter has a larger portion of tenderloin attached and comes from the rear end of the short loin; T-bones have less tenderloin, being cut from the front of the loin. Point is, when you're shopping, be sure to check the prices—odds are "boneless T-bone", "boneless porterhouse" and "New York strip" are the same cut of meat so be sure to get the best price. Gotta admire the marketing there.
Heating the cast iron in the oven first ensures even heat throughout the pan so, by getting it screaming hot on the stove afterwards, guarantees you a good, hard sear on the steaks. Feel free to skip moving the steak to the oven once flipped if your steak is less than 1" thick—just throw the butter in the pan and baste the meat with it a bit.
Don't get too hung up on the exact cut and thickness of beef; if you can't find a thinner strip steak as in the recipe, roll with the punches. The timing depends on the cut of meat, its thickness, and whether or not it has a bone in. For a New York strip (boneless) that's, let say, about 1" thick I'd figure 3 - 4 minutes on the stove and another 4 - 5 in the oven; something bone-in is going to take longer and should probably be in a slightly lower oven (400° - 425°).
If you're going for broke and doing an extravagant 2" thick T-bone, I'd go with the reverse sear method, similar to the one outlined here, adjusting the timing obviously.
BEEF INTERNAL TEMPERATURES—these are my guidelines
Blue……………………. 105 – 110° (remove at 100°-105° and rest for 5 minutes)
Rare………................... 125° (remove at 120° and rest for 5 minutes)
Medium-Rare………. 130°-135° (I usually remove at 125° and rest for 5 minutes)
Medium……………….. 135°-140° (I usually remove at 130° and rest for 5 minutes)
Medium-Well……….. 140°-145° (remove at 140° and rest for 5 minutes)
Well Done…………….. 155° (remove at 150° and rest for 5 minutes or not – its shoe leather anyway)