Heat oil in a large cast-iron skillet over medium. Arrange chicken, skin side down, in a single layer; cook, undisturbed, until skin is deeply golden and fat has rendered, 15 to 18 minutes. Turn chicken; cook, flesh side down, undisturbed, for 2 minutes. Transfer chicken to a plate. (Chicken will not be cooked through).

While chicken is rendering, separate dark green parts of scallions from white and pale green parts; thinly slice both and set aside separately.

Remove skillet from heat and pour off all but 1 tablespoon of the drippings; discard remaining drippings. Add garlic, ginger, 1 tablespoon butter, and reserved sliced white and pale green scallions to drippings in skillet. Cook over medium, stirring often, until fragrant and lightly golden, about 1 minute. Add mirin; cook, stirring constantly and scraping up any browned bits from bottom of skillet, until mirin is slightly reduced, about 2 minutes. Stir in stock, shoyu, and brown sugar. Bring to a simmer over medium-high.

Return chicken, skin side up, to skillet. Reduce heat to medium-low; cover partially. Simmer gently, undisturbed, over medium-low, for 10 minutes. Flip chicken; cook, uncovered, until a thermometer inserted into thickest portion of chicken registers 175°F, about 5 minutes. Transfer chicken, skin side up, to a clean large plate. Increase heat to medium-high; cook shoyu mixture, stirring often, until thickened and reduced to about 1 cup, 5 to 8 minutes. Remove from heat and gradually stir in remaining 5 tablespoons butter until sauce is emulsified. Return chicken, skin side up, to skillet and spoon some sauce over chicken. Garnish with remaining sliced dark green scallions and serve immediately.