In multi-task Bayesian optimization, the goal is to leverage experience from optimizing existing tasks to improve the efficiency of optimizing new ones. While approaches using multi-task Gaussian processes or deep kernel transfer exist, the performance improvement is marginal when scaling beyond a moderate number of tasks. We introduce BOLT, an initialization-only transfer strategy that distills prior BO runs into an LLM which proposes candidates for new tasks, while the surrogate at test time remains single-task. The LLM is periodically fine-tuned on top solutions from completed runs, creating a closed loop where better BO outputs yield better initializations over time. This decoupled design scales to roughly 1500 tasks without the saturation observed for shared-surrogate MTBO and adds only a small, amortized overhead relative to the BO inner loops. We evaluate on two domains: database query optimization and antimicrobial peptide design. We demonstrate that LLM-generated initializations steadily improve and accelerate BO, and with sufficient fine-tuning, a few LLM samples often match or surpass full ''from-scratch'' BO with far fewer oracle calls.