T-cells play a key role in adaptive immunity by mounting specific responses against diverse pathogens. An effective binding between T-cell receptors (TCRs) and pathogen-derived peptides presented on Major Histocompatibility Complexes (MHCs) mediate an immune response. However, predicting these interactions remains challenging due to limited functional data on T-cell reactivities. Here, we introduce a computational approach to predict TCR interactions with peptides presented on MHC class I alleles, and to design novel immunogenic peptides for specified TCR-MHC complexes. Our method leverages HERMES, a structure-based, physics-guided machine learning model trained on the protein universe to predict amino acid preferences based on local structural environments. Despite no direct training on TCR-pMHC data, the implicit physical reasoning in HERMES enables us to make accurate predictions of both TCR-pMHC binding affinities and T-cell activities across diverse viral epitopes and cancer neoantigens, achieving up to 72% correlation with experimental data. Leveraging our TCR recognition model, we develop a computational protocol for de novo design of immunogenic peptides. Through experimental validation in three TCR-MHC systems targeting viral and cancer peptides, we demonstrate that our designs-with up to five substitutions from the native sequence-activate T-cells at success rates of up to 50%. Lastly, we use our generative framework to quantify the diversity of the peptide recognition landscape for various TCR-MHC complexes, offering key insights into T-cell specificity in both humans and mice. Our approach provides a platform for immunogenic peptide and neoantigen design, opening new computational paths for T-cell vaccine development against viruses and cancer.