A persistent public health challenge is finding immunization schemes that are effective in combating highly mutable pathogens such as HIV and influenza viruses. To address this, we analyze a simplified model of affinity maturation, the Darwinian evolutionary process B cells undergo during immunization. The vaccination protocol dictates selection forces that steer affinity maturation to generate antibodies. We focus on determining the optimal selection forces exerted by a generic time-dependent vaccination protocol to maximize production of broadly neutralizing antibodies (bnAbs) that can protect against a broad spectrum of pathogen strains. The model lends itself to a path integral representation and operator approximations within a mean-field limit, providing guiding principles for optimizing time-dependent vaccine-induced selection forces to enhance bnAb generation. We compare our analytical mean-field results with the outcomes of stochastic simulations and discuss their similarities and differences.