Despite quantum theory's remarkable success at predicting the (statistical) results of experiments, many philosophers worry that it nonetheless lacks some crucial connection between theory and experiment. Such worries are at the root of the Quantum Measurement Problem. We can identify two kinds of worries: 1) pragmatic: it's unclear how to model our experiments to extract theoretical predictions, and 2) realist: there is no realist narrative for the experiment underlying these theoretical predictions. While both worries deserve attention, the pragmatic worries have far worse consequences if left unanswered. Moreover, as I will argue, upon reflection, a satisfactory explanation of almost all of quantum theory's experimental successes unavoidably involves modeling quantum fields at some point. Thus, without a pragmatic theory-to-experiment link for QFT, we are at risk of losing any right to claim evidential support for large parts of quantum theory. Hence, I focus on the Pragmatic QFT Measurement Problem. But, what makes modeling measurements in QFT so hard? As I will discuss, attempts to naively transplant our non-relativistic quantum measurement theory into QFT are deeply unphysical and unsatisfying. Thus we need a new (or at least refined) measurement theory for QFT. However, as I will argue, aiming too directly at a new measurement theory is an incautious way to proceed and is apt to lead us astray. This paper proposes an alternate way forward: We ought to first better understand how our non-relativistic quantum measurement theory is rooted in notions of measurement chains and Heisenberg cuts. Then we ought to generalize these notions and transplant them into QFT. Such a transplant is carried out in this paper. My analysis suggests the need for a pragmatic QFT-cut analogous to the need for a pragmatic Heisenberg cut in non-relativistic contexts.