If serving the stir fry with rice, quinoa, or other grain, prepare as needed. Drain and rinse the chickpeas, then spread on a paper or dishtowel and lightly pat dry. Set aside.
In a medium bowl, whisk all of the sauce ingredients together: water, soy sauce, hoisin, jam, rice vinegar, ginger, cornstarch, and 1 tablespoon chili garlic sauce. Taste and add additional garlic chili sauce if you prefer more heat.
Heat oil in a large, non-stick skillet or wok over medium high. Add the broccoli and bell peppers. Cook 2 minutes, then stir in the garlic, white parts of the green onions, and chickpeas. Cook for 3 additional minutes, until the vegetables are crisp tender, then pour in the sauce. Bring to a simmer and cook until the sauce thickens, about 2 additional minutes. Sprinkle with the green portion of the green onions and sesame seeds. Serve hot with prepared rice or quinoa.