We have developed a prototype of the photomultiplier tube (PMT) readout system for the Cherenkov Telescope Array (CTA) Large Size Telescope (LST). Two thousand PMTs along with their readout systems are arranged on the focal plane of each telescope, with one readout system per 7-PMT cluster. The Cherenkov light pulses generated by the air showers are detected by the PMTs and amplified in a compact, low noise and wide dynamic range gain block. The output of this block is then digitized at a sampling rate of the order of GHz using the Domino Ring Sampler DRS4, an analog memory ASIC developed at Paul Scherrer Institute. The sampler has 1,024 capacitors per channel and four channels are cascaded for increased depth. After a trigger is generated in the system, the charges stored in the capacitors are digitized by an external slow sampling ADC and then transmitted via Gigabit Ethernet. An onboard FPGA controls the DRS4, trigger threshold, and Ethernet transfer. In addition, the control and monitoring of the Cockcroft-Walton circuit that provides high voltage for the 7-PMT cluster are performed by the same FPGA. A prototype named Dragon has been developed that has successfully sampled PMT signals at a rate of 2 GHz, and generated single photoelectron spectra.