Edge-based and IoT devices have seen phenomenal growth in recent years, driven by the surge in demand for emerging applications that leverage machine learning models, such as deep neural networks (DNNs). However, a primary drawback of DNNs is their substantial storage/memory needs and high computational overhead, making their adoption in edge devices challenging. This limitation prompted the development of early-exit models like BranchyNet, which enable decisions to be made at earlier stages by incorporating dedicated exits within the architecture's inner layers. Nonetheless, these existing early-exit models lack control over the specific class that should exit and when. The necessity for such class-aware models is evident in numerous edge applications, where particular high-priority classes must be detected earlier due to their time-sensitive nature. In this article, we introduce ClassyNet, the first early-exit architecture designed to return only selected classes at each exit. This feature facilitates faster inference times for critical classes, allowing the initial layers to operate on edge devices. This strategy conserves considerable computational time and resources on the edge without compromising accuracy. Through extensive experiments, we show the effectiveness of ClassyNet compared to other models under various scenarios.