A/CMan is correct, your bullet is one of several variations for the 1850s/60s Sharps Rifle/Carbine. At the time of its introduction, it was a "revolutionary" advance in firearms technology, as one of the first successful Breechloading firearms. Breechloading means the bullet is loaded into the back end of the gun's barrel istead of at the front (muzzle) end. This resulted in much faster loading, and thus a faster firing rate, which is of course very important in Military service.
That being said, the Sharps Rifle was originally marketed to Civilian game-hunters more than to the US Army. But with the arrival of the civil war in 1861, the US Army purchased vast numbers of Sharps rifles and carbines, to give its troops an significant advantage over the Confederates who posssessed very few of the newfangled fast loading/firing Breechloaders. The Sharps was issued manly to cavalry, because loading firearm on horseback is more difficult than when standing on the ground. After the war ended, the size of the US Army shrank back to its pre-war numbers, so tens-of-thousands of its Sharps rifles were sold cheaply as "war surplus" to the civilian population.
Because you found your Sharps bullet in Vermont, it was almost certainly fired by a civilian game-hunter.
The great majority of Sharps Rifle bullets were .52-caliber, but there was also a (rare) .56-caliber version. If you can measure your bullet with digital calipers and tell us its diameter in hundredths-of-an-inch, we can tell you which version you found.
Ivan is correct that most of the 100+years-old lead bullets we dig out of the ground have developed a white or light grey Lead-Oxide patina due to "weathering." But there is a rare exception to that rule. When the lead bullet is buried in a "wet" environment, the water tends to prevent much Oxidation from happening. That MAY be why the bullet you found "on the shoreline of a lake" has so little white Lead-Oxide patina on it. I myself have dug 100%-sure civil war era bullets from creeks and swamp-muck which look like yours.