The only time I feed pinky mice is to adults who have been laying or breeding, to just give them an extra "boost" during an active breeding season. Even then, it is not very often.
At 10 1/2 inches, there is NO way your Beardie is big enough to be eating a pinky mouse, and there is probably no need to feed them to him to begin with. Once he is full gorwn, there is probably no harm in feeding one very occasionally as a "treat" (though again, not necessary), but you will probably find that they will not eat them unless they are live and wiggling.
If you want to give him variety besides his normal diet, you could try offering a few waxworns, silkworms, or something like that on occasion.
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