Basic Mayonnaise Recipe
Put one egg in a blender with 1 1/2 tablespoon vinegar or lemon juice, 1/2 tsp dry mustard, 1/4 tsp salt and pepper. Blend for a few seconds then slowly pour oil in while blending until it thickens.
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The great thing about making your own mayonnaise other than it being more natural and healthier, is that you can make it anyway you like. There are many different kinds of mayonnaise you can make. This site has some mayonnaise recipes. I didn't look at them all and don't know what the ingredients call for but if it says sugar, just replace it with splenda.
It occurred to me that mayonnaise, being basically egg, oil and some kind of acid (to keep the egg from going bad) can be made sweet too. Not only can you have mayonnaise for regular uses like tuna/chicken/egg salad or dipping meats in (a terrific alternative to ketchup), you can have a sweet version for dipping fruit or other low carb treats in or topping for desserts. Just replace vinegar with lemon juice and leave out the pepper and dry mustard. You can experiment with any spice or herb you want. You could even try adding cocoa or a little baker's chocolate to it. Homemade mayonnaise is similar to pudding in appearance and texture. You could make it into dairy-free pudding.