1. Prep ingredients
Bring a large pot of salted water to a boil. Finely chop 2 teaspoons garlic. Peel and finely chop 1 tablespoon ginger. Trim scallions, then thinly slice. Trim stem ends from broccoli, then cut crown into ½-inch florets. Coarsely chop cilantro leaves and stems together. Pat chicken dry; season with salt and pepper.
2. Cook chicken & broccoli
Heat 1 tablespoon oil in a large skillet over medium-high. Add chicken and cook, without stirring, until well browned on the bottom, about 3 minutes. Add broccoli and cook, stirring occasionally, until chicken is cooked through and broccoli is crisp-tender, 2–3 minutes more. Transfer to a plate. Wipe out skillet.
3. Cook aromatics
Heat 1 tablespoon oil in same skillet over medium-high. Add chopped ginger, garlic, and scallions. Cook, stirring, until fragrant, 1–2 minutes. Transfer to a bowl.
4. Make caramel sauce
In same skillet, combine ¼ cup sugar and 2 tablespoons water. Cook over medium-high, without stirring, until sugar is melted. Continue to cook, swirling skillet occasionally, until sugar turns honey-colored, 2–5 minutes (watch closely). Add ¾ cup water and 2 tablespoons black-bean garlic sauce; swirl (don't stir) to combine (caramel may harden, but will melt over heat).