Ken's right. R-M405/U106 is estimated to have formed about 4,800 to 5,000 years ago, and there are currently 5,168 branches (or subclades, e.g., more refined haplogroups) below it in the haplotree hierarchy.
The most common branch under R-L151 is P312 (15,594 subbranches), but U106 comes in second place with two additional branches representing only 249 and 29 subbranches, respectively.
The usefulness of the 23andMe haplogroup to you right now is that it can disprove a hypothesis, i.e., if another descendant of the most recent patrilineal ancestor shows a haplogroup that is not R-U106 or one of its subclades, then he and your brother are not related in the genealogical timeframe...not for millennia, in fact. But as a form of positive evidence, it can't tell you much.
One sort of sideways trick--and to be clear, nowhere near as accurate as taking a Big Y test or whole genome sequencing--is to see if you can find a known patrilineal relative(s) who has taken a Big Y test. If you're confident in the paper trail and he's no more distant than a 3rd or 4th cousin, then his haplogroup will give you a good idea of your brother's and father's.
As an example only, let's say that a 3rd cousin has taken a Big Y test and is positive for FGC12376, deep under U106. Your brother may not have that exact variant because we're talking a shared 4g-grandfather and a mutation may have occurred. But the next step higher from FGC12376 is FGC5253 (plus several synonymous variants). Another step higher from there is FGC5254, and then FGC5264. If the paper trail is accurate, it's a virtual certainty that your brother would be positive for one of those SNPs.
YSEQ, in Germany, offers high quality, low cost, a la carte Y-SNP testing. They currently have over a quarter of a million specific SNPs they can test. Turns out that, on the list, are FGC5253, FGC5254, and FGC5264. It would require a new cheek-swab sample and some waiting time, but the cost is only US$18 per SNP. In our hypothetical case of a Big Y-tested 3rd cousin (to whom you and/or your brother should be an autosomal DNA match for some verification that the relationship is as expected), I think I'd opt to spend the $54 and order tests for all three SNPs. One risk is that the paper trail could have an error or NPE; if so, you're out of luck and the results will come back as all negative--the money and test was basically wasted. The other risk is ordering only a single SNP: if you constrain the relationship too much by looking only for the cousin's terminal SNP or next higher, your brother might not match there but would match one more level up.
This, of course, provides you no yDNA comparison/matching info as you'd get if your brother tests with FTDNA. But if you can find the right yDNA-tested cousin who's willing to collaborate, the selective SNP testing method is an inexpensive way to "back in" to it and leverage the cousin's data and matches.