Side-channel analysis (SCA) is a class of attacks on the physical implementation of a cipher, which enables the extraction of confidential key information by exploiting unintended leaks generated by a device. In recent years, researchers have observed that neural networks (NNs) can be utilized to perform highly effective SCA profiling, even against countermeasure-hardened targets. This study investigates a new approach to designing NNs for SCA, called neuroevolution to attack side-channel traces yielding convolutional neural networks (NASCTY-CNNs). This method is based on a genetic algorithm (GA) that evolves the architectural hyperparameters to automatically create CNNs for side-channel analysis. The findings of this research demonstrate that we can achieve performance results comparable to state-of-the-art methods when dealing with desynchronized leakages protected by masking techniques. This indicates that employing similar neuroevolutionary techniques could serve as a promising avenue for further exploration. Moreover, the similarities observed among the constructed neural networks shed light on how NASCTY effectively constructs architectures and addresses the implemented countermeasures.