This paper deals with the brachistochronic motion of a thin uniform disk rolling on a horizontal plane without slip. The problem is formulated and solved within the frame of the optimal control theory. The brachistochronic motion of the disk is controlled by three torques. The possibility of the realization of the brachistochronic motion found in presence of Coulomb dry friction forces is inspected. Also, the influence of values of the coefficient of dry friction on the structure of the extremal trajectory is analyzed. Two illustrative numerical examples are provided.