Galaxy mergers play a critical role in driving galaxy evolution, especially by transforming galaxy morphology, redistributing the gas around galaxies, triggering AGN activity and star formation. We present the Cosmic Owl, a galaxy merger at $z=1.14$, identified in the COSMOS field. Deep imaging and spectroscopy from JWST, ALMA, and VLA reveal a complex system of twin collisional ring galaxies, exhibiting a nearly identical morphology. The grism spectra from the JWST COSMOS-3D program confirm that both galaxies host an AGN. A bipolar radio jet from one AGN extends to strike the merging front. In addition, we detect a starburst at the merging front, characterized by luminous extended nebular line emission and a massive cold gas reservoir. This starburst is likely triggered by interstellar shocks induced by galaxy collision and the AGN jet. The twin ring structure of the Cosmic Owl requires further numerical simulations to clarify the precise conditions that lead to the formation of this rare morphology. This system exemplifies how shock-induced star formation, driven by galaxy collision or AGN jet, can act as a previously underappreciated yet crucial mechanism for efficient gas-to-star conversion and rapid stellar mass assembly in the early Universe.