We have developed a nitrogen-incorporated silicon oxide (SiOxNy) deposition process using plasma enhanced atomic layer deposition (PEALD) for the gate insulator of recessed-gate Al-GaN/GaN metal-insulator-semiconductor heterojunction field-effect transistors. The SiOxNy film deposited on a recessed GaN surface exhibited a breakdown field of 13.2 MV/cm and a conduction band offset of 3.37 eV, which are the highest values reported for GaN MIS structures to the best of our knowledge. The fabricated normally-off transistor exhibited very promising characteristics such as a threshold voltage of 2.2 V, a maximum drain current density of 428 mA/mm, and a breakdown voltage of 928 V.