How can I make my own mayonnaise properly?

The oil needs to be slowly added to the egg, vinegar, and lemon juice base while being vigorously whisked. Only add more when the previous amount has been fully incorporated. I generally add the oil 1/4 cup at a time, dripping it very slowly into the base. Once the initial 1/4 cup has been added and emulsified it becomes a lot easier to incorporate the rest, and you can add it slightly faster, but still 1/4 cup at a time.

This is vastly simpler in a food processor. I have a Cuisinart 7-Cup Food Processor which has a little white pusher piece with a drip-hole in it. You can simply add the oil into this 1/4 cup at a time, it will slowly drip into the mix. Use the metal blade attachment.

You can make mayo in a blender. This makes the emulsion a bit easier to form and it is faster, and you can make loads without a stand mixer.

When making mayo in a blender (or any way) make sure the ingredients are cold. I've found it a bit easier to work the emulsion if I start 2 egg yolks in a bowl and whisk. Dump those in a blender, add 2 (whole) eggs, and pulse.

Open the top and start drizzling the oil. Pulse every couple of seconds, long enough to incorporate the oil. Be careful not to overwork the emulsion. It can break if you blend it too much.

To rebuild the emulsion after it breaks, beat another egg yolk, then whisk the broken emulsion back into the egg yolk a bit at a time.

Take the egg out of the fridge in advance, or put it under a warm water tap for a bit, so that it is at room temperature. Basically you want it at roughly the same temperature as the oil.

• In a bowl, put the egg yoke (no white, throw it away or keep it for something else), one teaspoon of French mustard and some salt (if not sure don't put too much, you can add more later to taste). Whisk that.
• Add about a table spoon of oil. I would recommend vegie oil, not olive oil. Whisk it hard! You want the emulsion to start happening, i.e. it should look consistent everywhere, with no apparent bubbles of oil. It should take about 10 seconds.
• Then repeat by adding no more than one table spoon of oil and whisking hard again until fully incorporated.
• After about a quarter cup of that, the mayo should look creamy already but still light, you may now accelerate the process but putting a bit more oil every time, and there's no need to whisk as hard or as much in between, unless you start seeing unincorporated oil.
• I find that after about a cup of oil, the mayo is good to serve, and adding more oil just makes it thicker and thicker -- Fascinating!
• Add salt and pepper to taste. You may add some lemon juice as well, it's especially recommended if you want to keep it out in the open air for longer than just dinner time, as it reduces oxidization.

It's much faster to do (and more fun!) with a partner:

• One person holds the bowl and keeps whisking.
• The other slowly pours the oil in a thin fillet.

It shouldn't take longer than 2-3 minutes overall. You may use an electric hand mixer. But I don't see the point of using a food processor, which will take much longer to clean afterwards!