In a saucepan, add vinegar, along with minced shallots, tarragon sprig, bay leaf, and a few cracks of pepper.
Turn stove on over low heat and bring to boil.
Stir the saucepan—you'll want to reduce the vinegar by a little over half.
Turn off heat, and let the saucepan hang out while you add an egg yolk into a large bowl.
Strain the vinegar from the saucepan (a little over a tablespoon) into the egg yolk.
Separately, take a stick of butter and cut in two, then place both halves into a microwave-safe measuring cup.
Heat in microwave for 30 seconds, then 15 seconds at a time until the butter is melted.
Slowly drip this warm butter into the egg-vinegar bowl, and start whisking.
You're looking for a loose consistency, not too liquid-like, but also not mayonnaise.
Then add chopped tarragon and mix well.
The butter is already salted, so taste to see if it requires additional salt.
Add a few cracks of pepper.
Serve immediately. Doesn't really save in the fridge well, sadly.
