Heat 1 tablespoon of sesame oil in a large pot over medium heat. Stir in the chopped onion and garlic; cook and stir until the onion has softened and turned translucent, about 5 minutes.
Pour in the chicken broth, green onions, and remaining sesame oil. Bring to a boil over high heat, and season to taste with salt and pepper.
Reduce the heat to medium, and add the frozen pot stickers and stir-fry vegetables. Cook until the pot stickers float, about 8 minutes.
