How do we estimate utility functions? Throughout the course load of an economics student's undergrad we are exposed to the concept of cardinal utility via utility functions which give us a specific quantifiable number of "utilis" as a result of our inputs which represent consumer goods.

Practically speaking though, what type of data is used here and what methods are employed to accurately measure utility in a cardinal sense?
One proxy for cardinal utility would be people's willingness to pay (WTP) for a particular product. One can estimate WTP using consumption data (e.g. from Amazon/eBay purchases), or elicit WTP using economic experiments. 

Another (better?) way to estimate cardinal utility is through methods used in neuroeconomics. For example, one can monitor an individual's dopamine level/active brain regions while she makes decisions. To the extent that the neuro-transmitter/brain activities are associated with bodily (physiological) pleasure, which by the way is what Jeremy Bentham had in mind when he coined the term "util", we may assert that particular activities/decisions generate a measurable level of utility that is cardinal in the sense that it not only orders choice alternatives from least to most preferred, but also signifies the intensity of preference over each alternative.