While significant improvements have been made in recent years in terms of end-to-end automatic speech recognition (ASR) performance, such improvements were obtained through the use of very large neural networks, unfit for embedded use on edge devices. That being said, in this paper, we work on simplifying and compressing Transformer-based encoder-decoder architectures for the end-to-end ASR task. We empirically introduce a more compact Speech-Transformer by investigating the impact of discarding particular modules on the performance of the model. Moreover, we evaluate reducing the numerical precision of our network's weights and activations while maintaining the performance of the full-precision model. Our experiments show that we can reduce the number of parameters of the full-precision model and then further compress the model 4x by fully quantizing to 8-bit fixed point precision.