Start by combining the black pepper, white wine vinegar, tarragon sprigs, shallot, white wine, and fennel seeds (if using) in a small saucepan. Set the heat to medium and let it simmer gently until reduced by half. This should take about 5-10 minutes. The reduction will create a flavorful base for your sauce.

After reducing, strain the mixture through a fine sieve into a bowl. You want to leave behind the solids while keeping that flavorful liquid. This step is crucial; it will ensure your sauce remains silky and smooth.

Fill a saucepan with water and bring it to a gentle simmer. Place a heatproof bowl on top—this is your makeshift double boiler. It’s essential for preventing your eggs from scrambling while cooking.

Add the strained vinegar mixture to the bowl over the simmering water. Toss in the egg yolks and start whisking vigorously. This step is where the magic begins; it’s crucial to whip the yolks continuously until they thicken and become frothy, which takes about 5 minutes.

Once thickened, start adding the cubed butter piece by piece. Let each cube melt into the mixture before adding the next. You're looking to create a creamy emulsion. Don’t rush this; it’s what gives Bearnaise its luxurious texture!

After all the butter is incorporated, add salt, lemon zest, and freshly chopped tarragon. Stir well and taste. Adjust seasoning as needed. You want that perfect balance of saltiness and herbal notes.

Bearnaise sauce should be served immediately, but if you need to hold off on serving, keep it warm over low heat. It can also be stored in a covered container in the fridge for up to a day. Just be mindful of reheating it gently.