In QFT we work with Lagrangians which contain terms $m$ such that the relativistic relation $E^2 = p^2 + m^2$ is satisfied. By classical analogy $m$ is called the 'mass'. We note that due to the existence of the Higgs field, spontaneous symmetry breaking can explain why some fields have zero and others nonzero values of $m$.
Presumably this quantity happens also to be the same $m$ that one would measure by, say, whacking a field quantum with a baseball bat and then studying its trajectory. My question is: why should this be? I.e. how does the rest mass of special relativity emerge from the mass terms of quantum fields?
Even more lovely would be an interpretation of how the Higgs mechanism might be understood from this perspective, or an explanation of for what reason no such interpretation is possible/useful. But this is a secondary goal.