When described by the wave picture, photonic devices are generally non-Hermitian systems, because of the coupling or leakage to an environment. In the presence of asymmetry in addition to non-Hermiticity, resonances appear in pairs, forming nearly-degenerate pairs called chiral pairs. The wave functions of a chiral pair are highly non-orthogonal to each other and exhibit emission chirality, that is, the imbalance between clockwise (CW) and counter-clockwise (CCW) propagating wave components. The appearance of chiral pairs is one of remarkable features of a non-Hermitian photonic system with asymmetry. I will show CW and CCW rotational travelling lasing modes are generated by locking of highly non-orthogonal resonance modes of a chiral pair in a partially-chaotic asymmetric billiard laser. From the view-point of laser dynamics, I will explain that only the CW or CCW chiral mode becomes a stable lasing state at the exceptional point. It is interesting that the light intensities of CW and CCW lasing modes are same in spite of the fact that CW components are much less than CCW components in resonance modes of a chiral pair, but these initial conditions which converge to CW lasing modes are much less than those which converge to CCW lasing modes.