Vaccination is expected to reduce disease prevalence and to halt the spread of epidemics. But pathogen adaptation may erode the efficacy of vaccination and challenge our ability to control disease spread. Here we examine the influence of the speed of vaccination rollout on the overall risk of pathogen adaptation to vaccination. We extend the framework of evolutionary epidemiology theory to account for the different steps leading to adaptation to vaccines: (1) introduction of a vaccine-escape variant by mutation from an endemic wild-type pathogen, (2) invasion of this vaccine-escape variant in spite of the risk of early extinction, (3) spread and, eventually, fixation of the vaccine-escape variant in the pathogen population. We show that the risk of pathogen adaptation is maximal for intermediate speed of vaccination rollout. On the one hand, slower rollout decreases pathogen adaptation because selection is too weak to avoid early extinction of the new variant. On the other hand, faster rollout decreases pathogen adaptation because it reduces the influx of adaptive mutations. Hence, vaccinating faster is recommended to decrease both the number of cases and the likelihood of pathogen adaptation. We also show that pathogen adaptation is driven by its basic reproduction ratio, the efficacy of the vaccine and the effects of the vaccine-escape mutations on pathogen life-history traits. Accounting for the interplay between epidemiology, selection and genetic drift, our work clarifies the influence of vaccination policies on different steps of pathogen adaptation and allows us to anticipate the effects of public-health interventions on pathogen evolution.