We propose a constricting Control Barrier Function (CBF) framework for prescribed-time control of control-affine systems with input constraints. Given a system starting outside a target safe set, we construct a time-varying safety tube that shrinks from a relaxed set containing the initial condition to the target set at a user-specified deadline. Any controller rendering this tube forward invariant guarantees prescribed-time recovery by construction. The constriction schedule is bounded and tunable by design, in contrast to prescribed-time methods where control effort diverges near the deadline. Feasibility under input constraints reduces to a single verifiable condition on the constriction rate, yielding a closed-form minimum recovery time as a function of control authority and initial violation. The framework imposes a single affine constraint per timestep regardless of state dimension, scaling to settings where grid-based reachability methods are intractable. We validate on a 16-dimensional multi-agent system and a unicycle reach-avoid problem, demonstrating prescribed-time recovery with bounded control effort.