Phenology, the timing of cyclical plant life events such as leaf emergence and coloration, is crucial in the bio-climatic system. Climate change drives shifts in these phenological events, impacting ecosystems and the climate itself. Accurate phenology models are essential to predict the occurrence of these phases under changing climatic conditions. Existing methods include hypothesis-driven process models and data-driven statistical approaches. Process models account for dormancy stages and various phenology drivers, while statistical models typically rely on linear or traditional machine learning techniques. Research shows that process models often outperform statistical methods when predicting under climate conditions outside historical ranges, especially with climate change scenarios. However, deep learning approaches remain underexplored in climate phenology modeling. We introduce PhenoFormer, a neural architecture better suited than traditional statistical methods at predicting phenology under shift in climate data distribution, while also bringing significant improvements or performing on par to the best performing process-based models. Our numerical experiments on a 70-year dataset of 70,000 phenological observations from 9 woody species in Switzerland show that PhenoFormer outperforms traditional machine learning methods by an average of 13% R2 and 1.1 days RMSE for spring phenology, and 11% R2 and 0.7 days RMSE for autumn phenology, while matching or exceeding the best process-based models. Our results demonstrate that deep learning has the potential to be a valuable methodological tool for accurate climate-phenology prediction, and our PhenoFormer is a first promising step in improving phenological predictions before a complete understanding of the underlying physiological mechanisms is available.