As Barbara suggests, omitting an egg may well compromise the cake's texture. Egg yolks are brilliant emulsifiers of fat and water. Egg whites, which are largely protein, in combination with the protein in your flour, are the muscle, if you will, of a cake (or anything else). They'll help quite literally hold everything up. If you're going to skip a trip to the store (and with temps where I live diving steadily below zero now that the sun has vanished, I completely understand), I would suggest boosting the flour by about 10% to compensate for the lost protein. Take whatever your flour measurement is in either cups or ounces and multiply it by .10. I'd not add any more fat, as it is difficult to artificially duplicate the complex lipids (fats) in a yolk. However, I would also suggest boosting your liquid (milk?) by also 10% to compensate for the water you're lacking from the missing egg white. Great question - thanks!
If you have a good vegan cook book kicking about there should be some notes on egg replacement (like flax seed, apple sauce, bananas). You could try some of those.
Personally I don't like using flax as an egg replacement as it makes the cake heavy, but banana works well for me (although it changes the taste). Haven't tried apple sauce yet.
Since you already have one egg, I don't think it will make a huge difference to use one of these egg substitutes. If you do, let us know how it turns out.
Think about it this way--old brownie mixes used to say that for "cake-like brownies," add another egg. (Maybe they still say that; I just haven't looked at a brownie mix in decades.)
For an answer to your question, I consulted one of my favorite authorities, Shirley Corriher. (Cookwise and Bakewise). She says that yolks work as emulsifiers and add creamy texture. Whites both dry and leaven (enable the cake to rise). So cutting eggs in half won't ruin the cake, but it won't rise as high as it would have, and it won't be as rich and creamy.
I don't think it will(ruin it).But it may not come out 100% as expected.I would add an extra half spoon of butter and a couple spoons of milk...or buy more eggs.It depends if you're just hungry for cake or if you want the perfect cake.
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For banana she says, 1/2 a banana mashed up = 1 egg, or 3Tbs apple sauce = 1 egg.
Personally I don't like using flax as an egg replacement as it makes the cake heavy, but banana works well for me (although it changes the taste). Haven't tried apple sauce yet.
Since you already have one egg, I don't think it will make a huge difference to use one of these egg substitutes. If you do, let us know how it turns out.