"Mayonnaise made without eggs that is creamy and smooth."
Eggs are one of the most efficient sources of protein there is.
I would agree that factory farming of eggs has a terrible effect but it seems like on a small scale it wouldn't necessarily be true.
(this is called the feed conversion ratio)
Eating vegetarian gives the baseline for comparison, a conversion ratio of ~1:1 (I think it is better to not claim exactly 1, animals eat different things than humans and maybe do better with extracting value from fiber).
The details of a given process for making a lower compromise meat replacement would have to be considered to calculate the equivalent measure. I don't think it would even be 2:1, but that's just a guess.
Eggs aren't even one of the most efficient animal based sources of protein [1]. When comparing the energy input in terms of fossil fuels against the energy output in terms of protein broilers have a 4:1 ratio, while eggs have a ratio of 40:1. The same value for corn is 2.2:1. (which goes up to 3.1 when accounting for bioavailability).
Looking at total caloric energy input against output isn't any better (worse for the broiler, better for the eggs). We here get a 6:1 conversion factor for eggs, and 9:1 for broilers [2].
[1] Sustainability of meat-based and plant-based diets and the environment: http://ajcn.nutrition.org/content/78/3/660S.full [2] Land, irrigation water, greenhouse gas, and reactive nitrogen burdens of meat, eggs, and dairy production in the United States: http://www.pnas.org/content/111/33/11996/F4.expansion.html