I want to estimate the second moment of a distribution. I know the breakdown of the second moment into the mean-squared and variance: $\mathbb{E}[X^2] = (\mathbb{E}[X])^2 + var(X)$.
When I want to estimate the second moment, however, it seems dubious to say $\widehat{\mathbb{E}[X^2]} = (\widehat{\mathbb{E}[X]})^2 + \widehat{var(X)} = \bar{x}^2 + s^2$.
In particular, why not $\widehat{\mathbb{E}[X^2]} = \bar{x}^2 + \widehat{\sigma}_{MLE}^2?$
Both seem like they can be defended as estimators of the second moment, but it is not at all clear to me that they have the nice properties like maximum likelihood and/or unbiasedness that we often like in our estimators.
What are the typical ways of estimating the second moment?