It's not as much about 'they each' than about 'number of'. Moreover I will switch some of your tenses for clarity as otherwise the verb forms are just the same.
As the dictionary tells us:
The construction the number of + plural noun is used with a singular verb (as in the number of people affected remains small). Thus it is the noun number rather than the noun people which is taken to agree with the verb (and which is therefore functioning as the head noun). -ODO: number, usage
The tricky part with each on the other hand is that it can act in two ways.
Used to refer to every one of two or more people or things, regarded and identified separately:
[AS DETERMINER]: each battery is in a separate compartment
each one of us was asked what went on
[AS PRONOUN]: Derek had money from each of his five uncles
they each have their own personality - ODO each
They each have peeled a number of potatoes.
This rephrasing of your sentence and the dictionary example are the pronoun use of each.
As you can see they each is followed by plural verb (have), and singular noun (a number of potatoes / own personality). In the dictionary example each refers to the individual properties of the people that are referred by they.
Now you are trying to record these numbers individually.
The number of potatoes they each have peeled is still singular, thanks to the number of which is still the head of the damn thing. Therefore it is a valid sentence with a singular verb noun agreement.
The number of potatoes they each peeled was recorded. (I changed tense for clarity.)
You can also use each as determiner and use singular all over:
The number of potatoes each of us has peeled is recorded.