Exploring quantum electrodynamics in the most extreme conditions, where electron-positron pairs can emerge in the presence of a strong background field, is now becoming possible in Compton collisions between ultraintense lasers and energetic electrons. In the strong-field regime, the colliding electron emits $\gamma$ rays that decay into pairs in the strong laser field. While the combination of conventional accelerators and lasers of sufficient power poses significant challenges, laser-plasma accelerators offer a promising alternative for producing the required multi-GeV electron beams. To overcome the complexities of colliding these beams with another ultraintense laser pulse, we propose a novel scheme in which a single laser pulse both accelerates the electrons and collides with them after self-focusing in a dedicated plasma section and reflecting off a plasma mirror. The laser intensity boost in the plasma allows the quantum interaction parameter to be greatly increased. Using full-scale numerical simulations, we demonstrate that a single 100 J laser pulse can achieve a deep quantum regime with electric fields in the electron rest frame as high as $\chi_e\sim 5$ times the Schwinger critical field, resulting in the production of about 40 pC of positrons.
