For the sauce: Whisk together the chicken stock, mushroom stock, ketchup, bean paste, cornstarch, chili sauce, honey, light soy sauce, brown sugar and dark soy sauce in a bowl.
For the batter: Whisk together the eggs, potato starch, salt and pepper in a bowl.
For the chicken: Heat the oil in a large saucepan or fryer to 350 degrees F. Add the seltzer to the batter and whisk to combine. Dip the chicken in the batter, then deep-fry until light brown, 2 to 3 minutes. Drain on paper towels.
Heat a wok over high heat. Add 2 to 3 tablespoons of the oil to the wok. Add the garlic, chiles and broccoli and saute until tender, 1 to 2 minutes. Add 1/2 cup of the sauce and cook, tossing, until slightly reduced. Add the rice wine and chicken and cook, tossing, until coated.
Ladle out 1/2 cup of the remaining sauce on each of 4 plates. Top each with the chicken and vegetables and sprinkle with the scallions.