E-commerce platforms increasingly display clickable query suggestions alongside items in the user feed, enabling users to refine or expand their intent without manually reformulating queries. Existing approaches either mine suggestions from historical logs—limited to past behavior and blind to long-tail, personalized intents or rely on off-the-shelf LLMs whose lack of platform-specific knowledge yields fluent but generic queries disconnected from real click behavior. We propose EAGER (Enrich-and-Align Generative Query Recommendation), a two-stage framework for generating query suggestions from clicked items. In the enrichment stage, supervised fine-tuning (SFT) follows a four-stage curriculum that scales information richness (from item-only to user-conditioned) and reasoning depth (from direct to chain-of-thought). Each stage incorporates rationale augmentation, diversity regularization, and self-distillation. In the alignment stage, we post-train via GRPO with a hybrid reward of multiple rule-based business signals and a preference-aware click reward. Extensive offline experiments and online A/B test demonstrate the effectiveness of EAGER, which has been deployed in production at a major e-commerce platform.