Evolution Strategies (ES) is a class of powerful black-box optimisation methods that are highly parallelisable and can handle non-differentiable and noisy objectives. However, naïve ES becomes prohibitively expensive at scale on GPUs due to the low arithmetic intensity of batched matrix multiplications with unstructured random perturbations. We introduce Evolution Guided GeneRal Optimisation via Low-rank Learning (EGGROLL), which improves arithmetic intensity by structuring individual perturbations as rank-$r$ matrices, resulting in a hundredfold increase in training speed for billion-parameter models at large population sizes, achieving up to 91% of the throughput of pure batch inference. We provide a rigorous theoretical analysis of Gaussian ES for high-dimensional parameter objectives, investigating conditions needed for ES updates to converge in high dimensions. Our results reveal a linearising effect, and proving consistency between EGGROLL and ES as parameter dimension increases. Our experiments show that EGGROLL: (1) enables the stable pretraining of nonlinear recurrent language models that operate purely in integer datatypes, (2) is competitive with GRPO for post-training LLMs on reasoning tasks, and (3) does not compromise performance compared to ES in tabula rasa RL settings, despite being faster.