We report on the demonstration of metalorganic vapor phase epitaxy-regrown (MOVPE) ohmic contacts in an all MOVPE-grown $beta$-Ga$_2$O$_3$ metal-semiconductor field effect transistor (MESFET). The low-temperature (600$^{circ}$C) heavy (n$^{+}$) Si-doped regrown layers exhibit extremely high conductivity with sheet resistance of 73 $Omega$/$square$ and record low metal/n$^{+}$-Ga$_2$O$_3$ contact resistance of 80 m$Omega$.mm and specific contact resistivity of 8.3$times$10$^{-7}$ $Omega$.cm$^{2}$ were achieved. The fabricated MESFETs exhibit a maximum drain-to-source current of 130 mA/mm, a high I$_{ON}$/I$_{OFF}$ of $>$10$^{10}$ with a high power FOM of 25 MW/cm$^{2}$ were achieved without any field plates. Nanoparticle-assisted Raman thermometry, thermal modeling, and infrared thermography were performed to assess the device self-heating under the high current and power conditions. This demonstration shows the promise of MOVPE technique for the realization of high-performance lateral $beta$-Ga$_2$O$_3$ devices and also highlights the need for device-level thermal management.
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