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The cookie recipe calls for 2 large eggs and I only have medium. Is there a way to make up for the egg loss or substitutions?

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There’s a good chance that the difference of what will be perhaps half a tablespoon just won’t matter in the final product. You don’t say what exactly you are going to make, but if your recipe was so sensitive to minute inaccuracies, it would probably be weight-based for all ingredients, including the eggs.

If you really feel your batter is too dry, add a small amount of liquid (milk or whatever fits your recipe) and you should be fine.

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    @DeniseNieceyWaddell I am sorry if I offended you. That wasn’t a snippy remark in my answer, rather trying to put you at ease. Some recipes do start with weighing the eggs and use that value to calculate the other ingredients. And that needs the precision of a scale. Your question is very good and helpful for future readers. I didn’t check your location, but if you are measuring your flour with cups, the inherent variance of that makes the difference in the eggs irrelevant. I didn’t write that in the answer, because I didn’t know then what you were going to make.
    – Stephie ♦
    Jun 8, 2020 at 5:45
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    From the wikipedia link, it would seem that you'd need an extra medium egg for every five to ten large eggs in the recipe.
    – HAEM
    Jun 8, 2020 at 11:10
  • @HAEM this sounds like a decent answer if you can elaborate a bit more.
    – Mołot
    Jun 9, 2020 at 8:24
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    This is the correct answer (at least in the EU), the range of what "medium" and "large" eggs are is so huge that those measurements are very imprecise anyways
    – Hobbamok
    Jun 9, 2020 at 8:33
  • @HAEM is "extra medium" bigger or smaller than regular medium?
    – Alex M
    Jun 9, 2020 at 16:01
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Just reduce the amount of the other ingredients slightly. For cookies, mainly this just means a little less flour. A little more or less butter and sugar doesn't make much difference - you just don't want the mix too dry. Take out a spoon or two of flour from what you measure, and you should be fine.

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Although for many things, you can get away with some imprecise measurements, when baking the difference in moisture can be quite significant, so for that you may want to use fractional eggs.

It's worth mentioning that 'large' and 'medium' are relative to your country, so a US large egg is equivalent to a UK medium egg.

For the US, a large egg is around 2 oz, while a medium egg is 1.75 oz. So you would need 2 2/7th eggs, or just a bit more than 1/4th of an egg.

For the UK, we're looking at the difference between ~68g vs. ~58g, so about 2 1/3 eggs.

For Australia, we have a larger difference, as it's ~43g vs. ~52g, so you'd need roughly 2 2/5 eggs.

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I was asked to expand on a comment I made under Stephie's answer. They posted a link to a Wikipedia article listing the regulations for egg sizes in various regions.

In North America, the minimum size for medium eggs is around 50 grams, and the minimum size for large is around 57 grams, so only seven grams per egg. That means that if a recipe calls for eight large eggs, using medium eggs would leave you about one egg short. Therefore, for a North American recipe, you should add one medium egg for every eight large eggs the recipe calls for.

The European sizes are 53 and 63 grams, respectively, or ten grams of difference per egg. So, in European terms, you should add one egg for every six large eggs when substituting medium eggs for large.

So, in conclusion, unless you are making very large batches of cookies, you don't really need to worry about the size of eggs. This sort of arithmetic might be more useful in the case you are using small eggs instead of large ones.

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