Protein Language Models (PLMs) have emerged as performant and scalable tools for predicting the functional impact and clinical significance of protein-coding variants, but they still lag experimental accuracy. Here, we present a novel fine-tuning approach to improve the performance of PLMs with experimental maps of variant effects from Deep Mutational Scanning (DMS) assays using a Normalised Log-odds Ratio (NLR) head. We find consistent improvements in a held-out protein test set, and on independent DMS and clinical variant annotation benchmarks from ProteinGym and ClinVar. These findings demonstrate that DMS is a promising source of sequence diversity and supervised training data for improving the performance of PLMs for variant effect prediction.