Begin by preparing the ingredients. Ensure the butter is clarified and warm (about 32°C to 38°C) and the egg yolks are ready in a bowl.
In a heatproof bowl, combine the egg yolks with the Dijon mustard, white wine vinegar, and fresh lemon juice. Whisk well until the mixture becomes slightly frothy and thickens (about 2-3 minutes).
Place the bowl over a pot of gently simmering water, ensuring the bottom of the bowl does not touch the water. This creates a gentle steam bath that heats the egg yolks evenly.
Slowly begin to add the warm, clarified butter to the egg yolk mixture while constantly whisking. Pour the butter in a thin stream while continuing to whisk until the sauce is thick and creamy (about 5-7 minutes).
Once the sauce reaches the desired consistency, remove the bowl from the steam bath. Season the sauce with salt and white pepper to enhance the flavor.
For an extra touch, you can optionally mix in the finely chopped tarragon or chives. Stir gently to distribute the herbs evenly.
Serve the Dijon Hollandaise sauce immediately or keep it warm by placing the bowl over warm water until ready to serve.