Bring the water and vinegar to the boil in a saucepan. Peel and chop the onions in the saucepan. Bring to the boil and simmer until half the liquid remains. Strain out the liquid.
Put water in the small pot and start boiling.
Separate the yolks into a metal bowl and place the bowl over the boiling water to make a 'water bath'. This is also called double boiler method.
Beat the eggs. When they start to warm up, add the vinegar liquid a little by little. Whisk vigorously.
When the mixture is smooth, start adding the butter a little at a time. Whisk vigorously and when all the butter has been used, season the Béarnaise Sauce with dried tarragon, white pepper, sugar and salt.
Let the Béarnaise Sauce stand for a minute to thicken.