Mayonnaise is one of the first recipes I learned working in a restaurant in high-school. There are many paleo mayo recipes on the net and even one in George and Juli’s new cookbook The Paleo Kitchen: Finding Primal Joy in Modern Cooking.
Many use a quick method using an immersion blender, but I can’t seem to get that to work for me, so I’m sticking to my trusty food processor. You can use any oil really for the 2 cups in this recipe; even bacon grease, but it’s best if you only use something that flavorful for ½ cup to one of the cups and use something else for the remainder of the oil. In the old days I would use canola for its neutral taste, but now I know better. I do not recommend this for olive oil unless it's just the last 1/2 cup or so. The violence of the food processor tends to make olive oil bitter.
My favorite mayo recipe
1 tablespoons white wine vinegar
1 tablespoons lemon juice
1 egg yolk*
1 whole egg*
1 teaspoon fine grain salt
1 teaspoon dry mustard
1-2 shakes pepper sauce (optional)
2 cups oil (I like one cup Avocado and 1 of a lighter oil) divided
Be sure all ingredients are at room temperature.
Measure out the oil and put it in a squeeze bottle (preferred) or a measuring cup with a spout.
Add all ingredients except the oil to the work bowl of a food processor.
Pulse 5 times.
Turn processor on and add oil in a steady stream (start one drop at a time if possible, but go SLOW) until incorporated.
Keep at room temperature for at least 30 minutes, but 2 hours is better.
Refrigerate for at least 30 minutes before using.
Everyone says it'll keep for a week in the fridge, but your mileage may vary.
*RAW EGG WARNING
Use caution in consuming raw and lightly-cooked eggs due to the slight risk of Salmonella or other food-borne illness. To reduce this risk, we recommend you use only fresh, properly-refrigerated, clean, eggs with intact shells, and avoid contact between the yolks or whites and the outside of the shell.
To get more time before it goes off, you can pasteurize the mayo in a sous vide. After step 6 above, cover tightly and submerge in a sous vide at 57 °C (134.6 °F) for 2 hours. Then chill in an ice bath for 10 to 15 minutes and refrigerate.
Adapted from Alton Browns Party Mayonnaise