Ion computed tomography (iCT) is an imaging modality for the direct determination of the relative stopping power (RSP) distribution within a patient's body. Usually, this is done by estimating the path and energy loss of ions traversing the scanned volume via a tracking system and a separate residual energy detector. This study, on the other hand, introduces the first experimental study of a novel iCT approach based on time-of-flight (TOF) measurements, the so-called Sandwich TOF-iCT concept, which in contrast to any other iCT system, does not require a residual energy detector for the RSP determination. A small TOF-iCT demonstrator was built based on low gain avalanche diodes (LGAD), which are 4D-tracking detectors that allow to simultaneously measure the particle position and time-of-arrival with a precision better than 100um and 100ps, respectively. Using this demonstrator, the material and energy-dependent TOF was measured for several homogeneous PMMA slabs in order to calibrate the acquired TOF against the corresponding water equivalent thickness (WET). With this calibration, two proton radiographs (pRad) of a small aluminium stair phantom were recorded at MedAustron using 83 and 100.4MeV protons. Due to the simplified WET calibration models used in this very first experimental study of this novel approach, the difference between the measured and theoretical WET ranged between 37.09 and 51.12%. Nevertheless, the first TOF-based pRad was successfully recorded showing that LGADs are suitable detector candidates for TOF-iCT. While the system parameters and WET estimation algorithms require further optimization, this work was an important first step to realize Sandwich TOF-iCT. Due to its compact and cost-efficient design, Sandwich TOF-iCT has the potential to make iCT more feasible and attractive for clinical application, which, eventually, could enhance the treatment planning quality