Put 2 tablespoons vegetable oil in a large skillet over medium-high heat.
Slice the chicken into 1/2-inch strips or chop into bite-sized pieces.
Add the chicken to the skillet, sprinkle with salt and pepper, and cook undisturbed until it sizzles and releases easily from the pan, about 3 minutes. After that, stir occasionally until it loses its pink color and browns in places, another 5 to 10 minutes.
Peel 1 inch ginger and 2 cloves garlic; chop them together.
Trim and chop the scallions, keeping the white and light green parts separate from the dark green tops.
Rinse and trim the chard, keeping everything but the toughest stem ends. Cut the leaves across the stem into ribbons as wide or thin as you like.
When the chicken is done, add the ginger, garlic, and white and light green parts of the scallions. Cook, stirring, until fragrant, about a minute. Transfer the chicken mixture to a bowl.
Add 2 tablespoons vegetable oil to the skillet and turn the heat to high. Add the chard, a handful at a time if necessary to fit it in, and sprinkle with salt and pepper. Cook until the leaves soften and begin to brown slightly, 5 to 7 minutes.
Return the chicken mixture to the skillet; add 2 tablespoons soy sauce and 1/2 cup water and stir until most of the liquid evaporates before turning off the heat. Taste and adjust the seasoning. Garnish with the scallion greens and serve, passing sesame oil at the table.