In a medium bowl, whisk together the soy sauce, alcohol-free mirin-style seasoning, chicken broth or water, brown sugar, grated ginger, and minced garlic.
Add the chicken pieces to the sauce and toss until evenly coated. Marinate for at least 15 minutes at room temperature, or refrigerate for up to 30 minutes.
While the chicken marinates, steam or boil the green beans for 3 to 5 minutes, until bright green and crisp-tender. Drain thoroughly.
Toss the warm green beans with the sesame oil and toasted sesame seeds, then set them aside.
Heat a large nonstick skillet or wok over medium-high heat. Remove the chicken from the marinade, reserving the remaining liquid.
Arrange the chicken in a single layer in the hot skillet. Cook for 3 to 4 minutes per side, until lightly browned.
Pour the reserved marinade into the skillet. Bring it to a simmer and cook for 5 to 7 minutes, stirring occasionally, until the chicken is fully cooked and the sauce forms a glossy glaze.
Remove the chicken from the heat and allow it to cool slightly before packing.
Divide the cooked rice, teriyaki chicken, and sesame green beans evenly among four bento containers.
Garnish with sliced green onion, if desired. Allow all components to cool before sealing the containers. Serve warm or at room temperature.