Matchgate circuits realize free-fermion dynamics: they are efficiently classically simulable, yet cannot on their own generate the generic randomness required for universal computation or unitary design formation. We study a controlled route beyond this integrable limit by doping matchgate circuits with non-Gaussian gates—physically, the injection of fermionic interactions into an otherwise free system. Using the matchgate commutant framework, we obtain analytic control over unitary 2-design formation. For globally scrambled dynamics, the design problem maps exactly onto a classical birth–death Markov chain with an Ornstein–Uhlenbeck continuum limit, recasting the emergence of quantum randomness in terms of spectral gaps and mixing times and yielding rigorous bounds on the number of non-Gaussian gates needed for approximate 2-designs. These bounds hold for a broad class of parity-preserving non-Gaussian gates, independently of microscopic details, with numerics indicating that the same mechanism governs higher-order designs. Used as local building blocks in a glued-circuit architecture, they yield approximate parity-preserving 2-designs in polylogarithmic depth with a sparse non-Gaussian gate count, with implications for Page-like entanglement growth and fermionic classical-shadow protocols. Finally, locality reshapes this picture: in local brickwork dynamics, design formation is diffusion-limited and far slower. Our results establish doped matchgate circuits as a controlled, analytically tractable route from free fermions to interaction-generated quantum designs.