Source: FOSS Cooking (community recipes)
Mayonnaise is an emulsification of oil into egg — the lecithin in the yolk acts as an emulsifier, binding fat and water into a stable suspension. This demands patience and a measured addition of oil; rush it and the emulsion breaks, leaving you with separated, greasy slop. A hand blender makes this foolproof by forcing vigorous mechanical action from the start.
Combine the whole egg, two yolks, vinegar, lemon juice, pickle juice, mustard, and salt in a tall, narrow container — the geometry matters, as it keeps the blender head submerged and working on the same batch of liquid. For aioli, add your garlic now (two to four cloves depending on how aggressive you want it). Pour the oil into a separate jug and set it within arm's reach. Do not add the oil to the container yet.
Position the hand blender at the very bottom of the container and run it at full power without moving it. You are creating violent turbulence. After ten to fifteen seconds — when you see a pale, thick emulsion forming around the blender head — begin adding oil in a thin stream whilst keeping the blender running and stationary. The oil should drip, not pour. As the mixture thickens and becomes glossy, you can increase the oil's flow rate slightly, but never rush. If the mixture looks like it's about to break — becoming grainy or slick — stop adding oil and blend what you have for another five seconds to re-stabilise emulsification before continuing. This takes three to four minutes in total.
Once all the oil is incorporated, the mayonnaise should be thick, pale, and hold its shape on a spoon. Taste it. The acid should sing — if it tastes flat and oily, add another teaspoon of vinegar or lemon juice and pulse briefly to combine. Pickle juice adds a sharp acid-in-cooking|acidity and salt simultaneously; don't skip it. For aioli specifically, the raw garlic should be present but not burn the palate; if it dominates, thin the batch slightly with a splash of cold water and re-blend.
Store in a sealed jar in the fridge for up to a week. It will thicken further as it cools.
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