We consider the advection-diffusion equation on $\mathbb{T}^2$ with a Lipschitz and time-periodic velocity field that alternates between two piecewise linear shear flows. We prove enhanced dissipation on the timescale $|\log ν|$, where $ν$ is the diffusivity parameter. This is the optimal decay rate as $ν\to 0$ for uniformly-in-time Lipschitz velocity fields. We also establish exponential mixing for the $ν= 0$ problem.