Search queries are typically very short, which means they are often underspecified or have senses that the user did not think of. A broad latent query aspect is a set of keywords that succinctly represents one particular sense, or one particular information need, that can aid users in reformulating such queries. We extract such broad latent aspects from query reformulations found in historical search session logs. We propose a framework under which the problem of extracting such broad latent aspects reduces to that of optimizing a formal objective function under constraints on the total number of aspects the system can store, and the number of aspects that can be shown in response to any given query. We present algorithms to find a good set of aspects, and also to pick the best $k$ aspects matching any query. Empirical results on real-world search engine logs show significant gains over a strong baseline that uses single-keyword reformulations: a gain of $14\%$ and $23\%$ in terms of human-judged accuracy and click-through data respectively, and around $20\%$ in terms of consistency among aspects predicted for "similar" queries. This demonstrates both the importance of broad query aspects, and the efficacy of our algorithms for extracting them.