Pat the chicken dry. Whisk soy sauce, mirin, sake or water, brown sugar, ginger, and garlic in a small bowl.
Heat oil in a large skillet over medium-high heat. Cook chicken smooth-side down until browned, 5 to 6 minutes. Turn and cook 3 minutes more.
Pour the sauce into the skillet and bring to a gentle simmer. Turn the chicken to coat and cook until the thickest part reaches 165°F (74°C), about 4 to 6 minutes, depending on thickness.
For a thicker glaze, transfer the chicken to a plate. Stir the cornstarch slurry into the simmering sauce and cook 30 to 60 seconds until glossy, then return chicken and coat. Skip this step for a thinner traditional-style sauce.
Rest chicken for 3 minutes, slice, and spoon sauce over the top. Garnish with scallions and sesame seeds. Serve with rice.