Speech recognition is a critical task in the field of artificial intelligence (AI) and has witnessed remarkable advancements thanks to large and complex neural networks, whose training process typically requires massive amounts of labeled data and computationally intensive operations. An alternative paradigm, reservoir computing (RC), is energy efficient and is well adapted to implementation in physical substrates, but exhibits limitations in performance when compared with more resource-intensive machine learning algorithms. In this work, we address this challenge by investigating different architectures of interconnected reservoirs, all falling under the umbrella of deep RC (DRC). We propose a photonic-based deep reservoir computer and evaluate its effectiveness on different speech recognition tasks. We show specific design choices that aim to simplify the practical implementation of a reservoir computer while simultaneously achieving high-speed processing of high-dimensional audio signals. Overall, with the present work, we hope to help the advancement of low-power and high-performance neuromorphic hardware.