Heat 1 tablespoon vegetable or canola oil in a large frying pan (12 inches) over medium-high heat until shimmering. Meanwhile, pat 1 ½ pounds boneless, skinless chicken thighs dry with paper towels and season all over with 1 teaspoon kosher salt.

Add the chicken to the pan in a single layer and cook undisturbed until well-browned on the bottom, 4 to 6 minutes. Meanwhile, place ⅓ cup mirin, ⅓ cup sake, ¼ cup soy sauce or tamari, ¼ cup water, and 2 tablespoons packed light or dark brown sugar in a medium bowl and whisk until the sugar is dissolved. Transfer 2 tablespoons of the sauce to a small bowl, add 1 tablespoon cornstarch, and whisk until the cornstarch is suspended.

Flip the chicken. Pour the sauce without the cornstarch over the chicken and bring to a boil. Reduce the heat as needed to maintain a lively simmer. Simmer until the chicken is cooked through, 5 to 10 minutes more.

Reduce the heat to medium. Transfer the chicken with tongs to a clean cutting board. Whisk the cornstarch mixture again. While whisking the sauce constantly, pour in the cornstarch mixture and cook until thickened, about 30 seconds. Turn off the heat.

Cut the chicken crosswise into ½-inch-wide slices and place in a wide, shallow serving bowl; pour the sauce over the chicken. Garnish with 1 thinly sliced medium scallion and white sesame seeds if desired. Serve with cooked white rice with cooked broccoli.