I've been trying to fill in a couple gaps in my overall knowledge of South Asian history, and one period that I'm quite curious about is Mughal and Islamic rule in India. What sources do people in this sub recommend to gain a broad, but still rigorous, overview of the period from around 1200 AD to the arrival of the British?
If you mean up to and including the arrival of the British, you may be interested in William Dalrymple's White Mughals. It really reads itself.
Although now a bit older (hard to believe sometimes), John F. Richards's The Mughal Empire is still relatively useful, even though it's just over the last few centuries. It is however a volume from the Cambridge History of India repackaged, and that work in general will be among the most encyclopedic treatments of the era; there are specific volumes on subjects like Vijayanagara, for example. However, South Asian history is one of the most dynamic fields in terms of theoretical reconsideration and adventuresome writing, so you may want to look for more recent work and get away from the "monuments" like the Cambridge history.
Pearson India had a series on Indian history that had a strong first volume (to 1200) by Upinder Singh, and a 1200-1750 volume by Ahmed Farooqui (A Comprehensive History of Medieval India, 2011). I've only read snippets from it, and I'm not sure I like the presentation or find it to have the same clarity, but that is an option.