Light-matter interactions are at the heart of quantum electrodynamics and underpin modern photonic technologies. As we develop means to control the properties of light, matter and their interactions, intriguing new phenomena emerge. Using a designer polariton platform we have developed, we reveal a long sought after Bardeen-Cooper-Schrieffer like phase in a particle-hole-photon strongly-coupled system. Coupling two trapped polariton condensates through both coherent tunneling and incoherent dissipation, we form a model system of rich nonlinear dynamics where new, equidistant frequency lines emerge via the limit cycles at Hopft bifurcation. Using two-dimensional monolayer crystals with exceptionally strong light-matter interactions, we control the exciton-photon interactions from the incoherent limit to the coherent limit with simple mirrors and laser pulses, showing the promise of the system for photonic applications based on coherent light-matter interactions. Combining different monolayers to form atomically-thin heterostructures, we obtain a platform that allows versatile control over both the photon modes and matter excitations, where we create long-lived valley excitons, ultra-thin lasers, and moire-lattice induced hybrid dipolar excitons and polaritons.