We develop a new Markov chain on graph partitions that makes relatively global moves yet is computationally feasible to be used as the proposal in the Metropolis-Hastings method. Our resulting algorithm can be made reversible and able to sample from a specified measure on partitions. Both of these properties are critical to some important applications and computational Bayesian statistics in general. Our proposal chain modifies the recently developed method called Recombination (ReCom), which draws spanning trees on joined partitions and then randomly cuts them to repartition. We improve the computational efficiency by augmenting the state space from partitions to spanning forests. The extra information accelerates the computation of the forward and backward proposal probabilities. We demonstrate this method by sampling redistricting plans and find promising convergence results on several key observables of interest.