The transfer of big data is increasingly supported by dedicated channels in high-performance networks, where transport protocols play an important role in maximizing application-level throughput and link utilization. The performance of transport protocols largely depend on their control parameter settings, but it is prohibitively time consuming to conduct an exhaustive search in a large parameter space to find the best set of parameter values. We propose FastProf, a stochastic approximation-based transport profiler, to quickly determine the optimal operational zone of a given data transfer protocol/method over dedicated channels. We implement and test the proposed method using both emulations based on real-life performance measurements and experiments over physical connections with short (2ms) and long (380ms) delays. Both the emulation and experimental results show that FastProf significantly reduces the profiling overhead while achieving a comparable level of end-to-end throughput performance with the exhaustive search-based approach.