Many biological processes, from cell division to viral lysis, are triggered when an internal stochastic variable reaches a threshold. Here we introduce Branching under First-Passage Resetting, a general framework in which replication events arise endogenously from first-passage dynamics rather than from externally imposed lifetime clocks. We show that the resulting population dynamics obey an exact renewal equation linking single-trajectory first-passage statistics to the population growth rate. This mapping shows that, for fixed offspring number and fixed mean replication time, stochastic timing fluctuations necessarily enhance growth relative to a deterministic clock. When offspring yield depends on the first-passage time, however, fluctuations have non-trivial effects and expose a fundamental yield-delay trade-off: waiting longer can increase the number of descendants, but delays all future lineages. Our framework allows us to address this optimization problem analytically, and upon application to bacteriophage lysis, gives an optimal lysis time and growth rate consistent with empirical data.