MARGAH, Afghanistan — With the signature whoosh and crack of a rocket-propelled grenade, the local Taliban of northern Bermel district made their feelings perfectly clear. After three years without laying eyes on U.S., Afghan, or allied forces, the insurgents of this mountainous border community were saying, unambiguously, that they were not exactly pleased that the U.S. Army had paid them a visit.

The rocket fell short, exploding in the loose, jagged rocks beneath one of a pair of mountaintops occupied by two platoons from Fox Company, 2nd battalion, 506th Parachute Infantry Regiment. The...