Large language models are rapidly becoming infrastructural components in high-stakes institutional settings, including public administration, legal reasoning, and healthcare, where opacity is not merely inconvenient but institutionally and legally untenable. Existing approaches to explainability are predominantly post-hoc, offering unstable, non-contestable accounts that have no formal relationship to the reasoning process that produced the output. We argue that the problem is not the absence of explanation but the absence of structured reasoning in the first place. This paper makes the case for a fundamentally different architecture, which we call the Glassbox Framework, in which Bayesian networks serve as transparent, ante-hoc mediation layers for generative models. Bayesian networks encode domain knowledge, causal assumptions, and probabilistic dependencies before inference occurs, enabling auditable reasoning traces, uncertainty quantification, and contestable outputs. We characterise the architecture of this framework and ground it in a benefit eligibility scenario, identifying the foundational challenges spanning semantic alignment, dynamic model construction, probabilistic grounding, and human governance that must be solved to realise it at scale. By shifting from post-hoc explanation to ante-hoc probabilistic mediation, this work outlines a principled path toward AI systems that are not only powerful but fundamentally accountable.