Digital assistants have become ubiquitous in e-commerce applications, following the recent advancements in Information Retrieval (IR), Natural Language Processing (NLP) and Generative Artificial Intelligence (AI). However, customers are often unsure or unaware of how to effectively converse with these assistants to meet their shopping needs. In this work, we emphasize the importance of providing customers a fast, easy to use, and natural way to interact with conversational shopping assistants. We propose a framework that employs Large Language Models (LLMs) to automatically generate contextual, useful, answerable, fluent and diverse questions about products, via in-context learning and supervised fine-tuning. Recommending these questions to customers as helpful suggestions or hints to both start and continue a conversation can result in a smoother and faster shopping experience with reduced conversation overhead and friction. We perform extensive offline evaluations, and discuss in detail about potential customer impact, and the type, length and latency of our generated product questions if incorporated into a real-world shopping assistant.