The Gauss-Markov theorem considers "best" as "lowest mean square error (MSE)" and a recent version of the theorem shows OLS is not only BLUE but also BUE: https://www.ssc.wisc.edu/~bhansen/papers/gauss.pdf
When the "best" becomes "lowest mean absolute error (MAE)", is it the case that the best estimator is no longer OLS and perhaps quantile (median) regression?