Probably handling angry customers.这里handling是动名词
动名词与一般名词的不同在于具有动词特征 可以带自己的宾语.
angry是形容词 修饰customers.
angry customers是动名词的宾语.
handling angry customers是动名词短语,可以看作 是一个抽象名词.
本问题 不是一个句子 而是一个省略谓语的 答语.
如what was the most difficult part of the job?
Probably handling angry customers.
=Probably handling angry customers was the most difficult part of the job.
这是动名词短语作主语