If cooking rice, prepare it according to package directions and keep warm. If using microwaveable ready rice, wait and heat it just before serving.
Heat the oil in a large skillet or wok over MEDIUM-HIGH heat. Add the ground beef and cook, breaking it up with a spoon, until browned and cooked through, about 5 minutes. Drain excess grease if needed.
Stir in the ginger paste and garlic and cook for about 30 seconds, until fragrant.
Add the frozen stir-fry vegetables and cook uncovered for 4 to 6 minutes, stirring occasionally, until heated through. Continue cooking as needed to allow excess moisture to evaporate. If your stir-fry mix has large broccoli florets, they may take a little longer to cook. You can cover the pan for a minute or two to help them soften, then uncover and continue cooking until the sauce thickens.
In a small bowl, whisk together the teriyaki sauce and cornstarch until smooth. Pour the sauce into the skillet and stir to combine. Cook for 2 to 3 minutes, stirring, until the sauce thickens and coats the beef and vegetables. Garnish with green onions.
If using ready rice, heat it according to the package directions. Divide the rice among the bowls and spoon the teriyaki beef mixture over the top. Garnish with additional green onions and sesame seeds, if desired.