Markov Chain Monte Carlo (MCMC) underlies both statistical physics and combinatorial optimization, but mixes slowly near critical points and in rough landscapes. Parallel Tempering (PT) improves mixing by swapping replicas across temperatures, yet each replica still relies on slow local updates to change its configuration. We introduce IsingFormer, a Transformer trained on equilibrium samples that can generate entire spin configurations resembling those from the target distribution. These uncorrelated samples are used as proposals for global moves within a Metropolis step in PT, complementing the usual single-spin flips. On 2D Ising models (sampling), IsingFormer reproduces magnetization and free-energy curves and generalizes to unseen temperatures, including the critical region. Injecting even a single proposal sharply reduces equilibration time, replacing thousands of local updates. On 3D spin glasses (optimization), PT enhanced with IsingFormer finds substantially lower-energy states, demonstrating how global moves accelerate search in rugged landscapes. Finally, applied to integer factorization encoded as Ising problems, IsingFormer trained on a limited set of semiprimes transfers successfully to unseen semiprimes, boosting success rates beyond the training distribution. Since factorization is a canonical hard benchmark, this ability to generalize across instances highlights the potential of learning proposals that move beyond single problems to entire families of instances. The IsingFormer demonstrates that Monte Carlo methods can be systematically accelerated by neural proposals that capture global structure, yielding faster sampling and stronger performance in combinatorial optimization.