In this case I’m pretty certain that aside from its input circuits the internal signal path of the A21 is single-ended, but several things in its description, specs, and manual appear to indicate that its input circuits which receive the signals provided to its XLR connectors are in fact balanced and differential. Including a statement in its description that it provides "balanced inputs with discrete circuits and XLR connectors"; the fact that its input impedance is specified as "33 k Ω unbalanced; 66 k Ω balanced"; and various statements in the manual referring to the advantages of balanced interconnections.
Also, while it is true that connecting a balanced output to a balanced input (as you indicated you would be doing in connecting the Mytek DAC to the A21) will often result in 6 db more overall system gain than would be provided by an unbalanced connection between the same components, that is not always the case. It depends on the designs of the specific interface circuits in **both** components. And after looking at the descriptions and manuals of the two components I don’t see enough information to determine the answer in this case.
BTW, I see also that the Brooklyn DAC incorporates internal jumpers that are said to provide the capability of reducing its output levels by 6 db. That may not have any relevance to your question, but I thought I’d point it out.
Regards,
-- Al