To make the mayonnaise by hand: Rinse a mixing bowl with hot water and dry it well. Place the 2 egg yolks, vinegar, mustard, and a little salt and pepper in the bowl and beat the mixture with a whisk or an electric mixer at medium speed (3 min / speed 4 on TM5 ) until very well blended. Add ½ cup (125 ml) of the oil drop by drop, beating constantly. Continuing to beat, add the rest of the oil in a thin stream, being careful to incorporate all the oil to form an emulsion. When all the oil has been absorbed, season the mayonnaise to taste with salt, pepper, and lemon juice. Beat in a little broth or water if you wish to thin the mayonnaise. To make mayonnaise in a food processor or blender: Put 1 egg, 1 egg yolk, the vinegar, the mustard, a little salt, and a few grinds of pepper in the bowl of the processor fitted with the steel blade and blend well. With the processor running, add the oil in a thin, steady stream. When the mayonnaise is emulsified and all the oil has been absorbed, scrape it into a bowl and adjust the seasoning to taste with salt, pepper, and lemon juice. Beat in a little broth or water if you wish to thin the mayonnaise.
You can use an immersion blender too. If you do, put the ingredients in a jar, allow the oil to settle on top. Put the immersion blender flush to the bottom and make sure it is below the oil. Turn on and slowly pull the immersion blender up as the emulsified mayonnaise rises from the bottom. See: immersion blender video from Kenji Lopez or the recipe on Serious Eats.
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