The BLS does document their process in detail on their FAQ [0].
This might not match your experience because your family doesn't shop the same basket of goods the families the BLS surveyed do. There are also regional variations in the prices of goods. Even store prices in the same region vary. All of these variations get averaged out.
This is why you can't extrapolate from n=1.
All-items was 3.7%. That does not mean components with-in the overall rate were also 3.7%. Particular components may have gone up more that the overall rate, but only a fraction of that is part of the all-items that is seen in the headlines.
Further, if your personal basket of goods is most likely different than the average basket of goods mix, so your personal inflation rate will be different from the headline number. StatCan, who does the CPI for Canada, actually has a tool for finding out your personal inflation rate based on your purchases:
* https://www150.statcan.gc.ca/n1/pub/71-607-x/71-607-x2020015...
The idea that the government would cook the books in their favor isn't outside the realm of possibility. The idea that it's been happening for 40+ years is worrying.
Every admin, both major parties and all of big business gain from understating inflation, it's a fundamental conflict of interest. Calling it a "possibility" might be a tiny tad timid, I think.
There's lots of data out there, for example: https://prices.marketreportblog.com