We introduce a Bergman-space framework for the study of boundary-forced heat equations and show that, in the one-dimensional case, boundary white noise gives rise to a sharp holomorphic regularity phenomenon. More precisely, we consider the heat equation on a bounded interval with Dirichlet or Neumann boundary conditions driven by independent white noises at the endpoints, and we prove that for every positive time the corresponding state extends holomorphically to a rhombus in the complex plane having the original interval as one of its diagonals. Moreover, the resulting process admits a continuous version with values in a scale of weighted Bergman spaces on that rhombus, depending on two parameters $δ\in(0,1)$ and $Θ\in\left(0,\fracπ{4}\right)$. To our knowledge, this is the first systematic use of Bergman spaces as state spaces for parabolic equations with stochastic boundary forcing. We also prove that the result is optimal, in the sense that the conclusion fails at the critical values $δ=0$ and $Θ=\fracπ{4}$.