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I was looking for a book that could explain me well the Generalized Method of Moments, its mathematical nuances, and even have a look to the empirical side, maybe with some guided exercises with Stata or another software package. Thanks in advance!

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I second the choice of Hayashi. Additionally, you will find more detail (not on Stata, though) in Generalized Method of Moments, Alastair R. Hall.

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Econometrics by Fumio Hayashi is probably the book you are looking for. It treats a range of estimation techniques, such as OLS, IV etc., as special cases of GMM. A lot of the chapters include empirical applications of the methods used.

It does give data sets for the empirical exercises but does not cover any specific software package.

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