When you buy a bottle of single malt whisky, you are investing in a product that is made of dozens of different whiskies that are married to create the same taste year after year. The word "single" refers to the fact that all the whiskies come from one distillery; if there is an age statement on the label, this refers just to the youngest whisky in the mix. The upside of this process is that if you have a favorite, you can buy it over and over again. But if you want to try a whisky that is never mixed with something from another cask, you have to seek out releases from independent bottlers.
Independents buy "one-off" casks of whisky from big-name distillers—often climbing through warehouses in Scotland, sampling different barrels and picking out the very best—and bottle them under their own name or on behalf of a shop, restaurant or hotel. The best independent bottler that I know of is SIGNATORY, whose range includes a remarkable collection under the Vintage Cask label. It is difficult to recommend one because when it's gone, it's gone. But a trawl on the Internet will reveal American stores that have bottles (try totalbeveragesolution.com). At the time I wrote this, I found a Signatory Macallan 1990 ($55) and the quite wonderful Signatory Brora 1981 ($130). Set a price point, then take a flier. It's the most exciting way to approach malt whisky.