Large language models (LLMs) capable of casual conversation have recently become widely available. We hypothesize that users of conversational systems want a more personalized experience, and existing work shows that users are highly receptive to personalized questions (PQs). Question Generation tasks, however, focus on factual questions from textual excerpts. To create a PQ generator, we first identify over 400 real user interests by anonymously aggregating ~39K user models. We then populate prompt templates with these 400 interests and use an LLM to generate PQs customized to user interests. The result is PerQs, a novel corpus of ~19K question/answer pairs. We evaluate PerQs at scale in the unique context of the Alexa Prize. Our results show significant positive effects on perceived conversation quality. We then fine-tune, deploy, and evaluate PerQy, a neural model that generates PQs in real-time. When evaluated against several competitive LLM baselines, PerQy produced the most natural and engaging responses.