Combine broth, sherry, brown sugar, 2 teaspoons each of garlic and ginger, soy sauce, vinegar, lemon juice, and pepper flakes in a bowl.
Sear chicken breasts in 1 teaspoons oil in a nonstick skillet over medium-high heat. Cook until deeply brown on both sides, about 4 minutes per side. Remove chicken from pan.
Deglaze pan with broth mixture and simmer until sauce is reduced to a syrup, about 5 minutes more. Return chicken to pan to coat with sauce. Transfer chicken and sauce (there will only be a little sauce) to dinner plates; tent to keep warm.
Wipe skillet with a paper towel.
Stir-fry broccoli, bell pepper, mushrooms, and onion with remaining garlic and ginger in remaining 1 teaspoons oil in the skillet over medium-high heat. Cook until vegetables are barely tender, 4-5 minutes. Turn off heat and stir in sprouts.