Head and neck cancers often emerge within genetically altered fields of premalignant cells that appear histologically normal but have a high chance of progression to malignancy. Clinical consequences of this so-called field cancerization are multifocal lesions and high recurrence risks after excision of the primary tumor. We develop a spatiotemporal stochastic model of head and neck tumorigenesis, combining evolutionary dynamics at the phenotypic level with a general framework for multi-stage progression to cancer. Based on the model, we derive probabilistic distributions for clinically relevant quantities such as size of the invisible premalignant field at time of cancer diagnosis, and risk of local and distant recurrences. Finally, we discuss how our model can be combined with patient-specific measurements to optimize surgical excision margins and post-operative monitoring.