Boolean circuits abstract away from physical implementation details to capture the logical structure and computational behaviour of digital components. Despite their long history of study, their compositional foundations have received comparatively little attention. Building on the theoretical foundations of computons introduced in previous work, we develop a concrete and operationally defined model of computation for control-driven Boolean circuits. We introduce colimit-based operators for composing circuits into sequential, parallel, branching and iterative structures, supported by concrete operational semantics grounded in particular abstract computing devices. The framework refines the semantics of composition, establishes the algebraic properties of its operators and introduces atomic circuits to support more expressive computation units. We demonstrate the applicability of the proposed framework through examples from propositional logic, biomedicine and electronics, and establish an operational equivalence with respect to classical Boolean circuits through a systematic transformation that accommodates gates of arbitrary arity. By this equivalence, we show that control-driven Boolean circuits are at least as powerful as their classical counterpart. That is, they are capable of non-uniformly computing any Boolean function over inputs of arbitrary length. Implementations in Idris and Java demonstrate the practical realisability of the framework.

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