We report the growth of high quality Bi$_{2-x}$Sb$_x$Te$_{3-y}$Se$_y$ ultrathin nanoplates (BSTS-NPs) on an electrically insulating fluorophlogopite mica substrate using a catalyst-free vapor solid method. Under an optimized pressure and suitable Ar gas flow rate, we control the thickness, the size and the composition of BSTS-NPs. Raman spectra showing systematic change indicate that the thicknesses and compositions of BSTS-NPs are indeed accurately controlled. Electrical transport demonstrates that a robust Dirac cone carrier transport in BSTS-NPs. Since BSTS-NPs provide superior dominant surface transport of the tunable Dirac cone surface states with negligible contribution of the conduction of the bulk states, BSTS-NPs provide an ideal platform to explore intrinsic physical phenomena as well as technological applications of 3-dimensional topological insulators in the future.