Depending on the problem you may use Bayesian Neural Networks where you fit a distribution over the weights they are better calibrated but also expensive. There exists some theory on lower cost ways to make the model better calibrated / uncertainty aware. One direction is using Gaussian Process approximations an other is for example PostNet. The overal topic you can search for is uncertainty quantification

By any chance do you have the name or a reference?

XGBOOST is not strictly better than NNs in classification. I can guarantee you as I worked on a classification dataset where a NN performed significantly better then XGBOOST. While your statement will be correct for small sample datasets I am pretty sure, that it will not hold for large datasets with complicated features