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1
Fill a large saucepan a quarter of the way full with water and bring it to a simmer over high heat.
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2
Remove from the heat and set aside.
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3
Combine the wine, vinegar, shallot, tarragon stems, and crushed peppercorns in a medium frying pan.
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4
Bring to a simmer over medium heat and cook until reduced to just over 2 tablespoons, about 5 to 7 minutes.
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5
Strain the vinegar mixture through a fine-mesh strainer into a small bowl, pressing on the solids; discard the solids.
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6
Set aside to cool, about 10 minutes.
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7
Place the egg yolks and the cooled vinegar reduction in a blender and blend on medium speed until the mixture lightens in color, about 30 seconds.
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8
Reduce the blender speed to low, remove the small cap from the lid (the pour lid), and slowly drizzle in all of the melted butter.
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9
Turn off the blender; add the chopped tarragon leaves, chervil (if using), salt, and pepper; and pulse to combine.
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10
To keep the bearnaise sauce warm, place the blender pitcher directly in the reserved hot water while you cook the steak and poach the eggs.
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11
Heat a medium heavy-bottomed frying pan (not nonstick) over high heat until very hot but not smoking, about 3 to 4 minutes.
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12
(If the pan gets too hot and starts to smoke, take it off the heat to cool a bit.)
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13
Meanwhile, season the steak on both sides with the measured salt and pepper.
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14
Rub both sides with the olive oil.
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15
Place the steak in the hot pan and let it cook undisturbed until a dark crust forms on the bottom, about 3 to 4 minutes.
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16
Flip the steak using tongs or a spatula and cook until its medium rare, about 3 to 4 minutes more.
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17
To check for doneness, use your finger to press on the steak: It should be firm around the edges but still give in the center.
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18
You can also use an instant-read thermometerit should read 125 degrees F to 130 degrees F.
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19
Transfer the steak to a cutting board and let it rest while you poach the eggs.
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20
Bring the measured water to a simmer in a medium pot over high heat (the water should be at least 2 inches deep).
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21
Reduce the heat to low to keep the water at a bare simmer.
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22
Stir in the vinegar.
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23
Break 1 egg into a small cup or ramekin.
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24
Gently slide the egg into the simmering water.
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25
Repeat with the remaining 3 eggs, making sure to evenly space them in the water.
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26
Cook until the whites are just set, about 3 minutes.
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27
Lift the eggs out of the water with a slotted spoon.
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28
Place in a bowl of warm water to keep warm.
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29
Split and toast the English muffins.
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30
Coat the cut sides of the muffins with a thin layer of butter, season with salt and pepper, and place on serving plates.
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31
Thinly slice the steak and divide it among the muffin halves.
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32
Using a slotted spoon, remove the eggs from the warm water.
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33
Blot any excess water from the underside of the spoon with a paper towel or kitchen towel and slide the eggs onto the steak.
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34
Top the eggs with the desired amount of bearnaise sauce (you may have some left over to serve on the side).
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35
Serve immediately.
