We investigate the mutual interaction of two spatially-separated Airy beams in the nonlinear Schrödinger equation with the fractional Laplacian. Depending on the beam separation (d), relative phase and Lévy index (α), we observed an anomalous attraction or repulsion between the Airy beams. Anomalous attraction leads to a single breather soliton with a period that grows exponentially as α increases. In this region of the parameter space, we identify a crossover between two asymmetric regimes: as the Lévy index exceeds a critical value α _c, the period of breather soliton for d>0 is orders of magnitude larger than for d<0, while the opposite occurs as α<α _c. Our results reveal a novel scenario for Airy beams interaction in the framework of fractional nonlinear Schrödinger equation and provide an alternative mechanism to control breather soliton generation.