Existing aspect extraction methods mostly rely on explicit or ground truth aspect information, or using data mining or machine learning approaches to extract aspects from implicit user feedback such as user reviews. It however remains under-explored how the extracted aspects can help generate more meaningful recommendations to the users. Meanwhile, existing research on aspect-based recommendations often relies on separate aspect extraction models or assumes the aspects are given, without accounting for the fact the optimal set of aspects could be dependent on the recommendation task at hand. In this work, we propose to combine aspect extraction together with aspect-based recommendations in an end-to-end manner, achieving the two goals together in a single framework. For the aspect extraction component, we leverage the recent advances in large language models and design a new prompt learning mechanism to generate aspects for the end recommendation task. For the aspect-based recommendation component, the extracted aspects are concatenated with the usual user and item features used by the recommendation model. The recommendation task mediates the learning of the user embeddings and item embeddings, which are used as soft prompts to generate aspects. Therefore, the extracted aspects are personalized and contextualized by the recommendation task. We showcase the effectiveness of our proposed method through extensive experiments on three industrial datasets, where our proposed framework significantly outperforms state-of-the-art baselines in both the personalized aspect extraction and aspect-based recommendation tasks. In particular, we demonstrate that it is necessary and beneficial to combine the learning of aspect extraction and aspect-based recommendation together. We also conduct extensive ablation studies to understand the contribution of each design component in our framework.