In this paper, we study a class of locally recoverable codes in which any two recovering sets of a coordinate position can intersect at a small number of elements, but the intersection of any three recovering sets of a coordinate is empty. This subclass of LRC codes with intersecting recovering sets is important because a code with t recovering sets in this subclass can handle $\left\lceil {\frac{t}{2}} \right\rceil$ simultaneous erasures locally while having all the properties of the parent class (LRC codes with intersecting recovering sets).