The dynamics of quantum many-body systems is intricately related to random matrix theory (RMT), to such a degree that quantum chaos is even defined through random matrix level statistics. However, exact results on this connection are typically precluded by the exponentially large Hilbert space. After a short introduction to the role of RMT in many-body dynamics, I will show how dual-unitary circuits present a minimal model of quantum chaos where this connection can be made rigorous. This will be illustrated using a new kind of emergent random matrix behaviour following a quantum quench: starting from a time-evolved state, an ensemble of pure states supported on a small subsystem can be generated by performing projective measurements on the remainder of the system, leading to a projected ensemble. In chaotic quantum systems it was conjectured that such projected ensembles become indistinguishable from the uniform Haar-random ensemble and lead to a quantum state design, which can be shown to hold exactly in dual-unitary circuit dynamics.