Knowledge distillation (KD) is one of the most effective paradigms for compressing large-scale foundation models into deployable architectures. In the context of Automatic Speech Recognition (ASR), previous studies have predominantly focused on forcing the student model to strictly mimic the predictive distribution of a massive teacher model. However, this static dependency often presents an inherent trade-off: while the student rapidly acquires basic linguistic representations, it simultaneously inherits the teacher's domain-specific blind spots and over-confident hallucinations, leading to a severe decline in out-of-distribution generalization capacity. To effectively mitigate this issue, we propose Adaptive Self-Knowledge Distillation (ASKD), a dynamic curriculum framework. ASKD systematically decays the dependency on the teacher's distribution as training progresses-thereby unlocking the student's independent reasoning capacity-and subsequently employs a self-knowledge distillation phase to act as a structural regularizer. By applying ASKD, we distill the massive Whisper architecture into a compact variant, ASKD-Whisper. In our comprehensive evaluations across diverse acoustic domains, ASKD-Whisper not only achieves a 5x speedup in inference latency but also outperforms its teacher model by yielding a 1.07% lower word error rate (WER). These results demonstrate that ASKD effectively prevents teacher-induced overfitting and establishes a new state-of-the-art for generalizable model compression.