We have developed a method to fabricate graphene flakes composed of high quality multi-layer graphene sheets using a thermal plasma jet system. A carbon atomic beam was generated by injecting ethanol into Ar plasma continuously; the beam then flowed through a carbon tube attached to the anode. Graphene was made by epitaxial growth where a carbon atomic beam, having the proper energy, collided with a graphite plate. The graphene fabricated was very pure and showed a relatively good crystalline structure. We have demonstrated that the number of layers of graphene sheets could be controlled by controlling the rate of ethanol injection. Our process is a continuous process with a relatively high yield (approximately 8%).