Cook broccoli: Bring a pot of water to a boil and blanch the broccoli florets for 2 minutes. Drain in a colander and rinse with cold water. Let drain very well. Pat the broccoli dry with kitchen paper. This step is not entirely necessary, but it is recommendable. Too much water would thin down the sauce.
Dry tofu: Place the tofu between two layers of kitchen paper and press down with the hand to absorb as much moisture as possible. Repeat the procedure a couple of times with new kitchen paper. Finally, cut the tofu into 2.5 cm/1-inch cubes and set them aside.
Sauce: whisk the tomato paste, ketchup, soy sauce, rice vinegar, hoisin sauce, chili sauce, sesame oil, and sugar. Set aside.
Chop: Finely mince the dried chilies, the garlic cloves, and the ginger. Slice the scallions into thin rings, keeping the white and the green parts separated. Set aside.
Cook tofu: Heat 1 tablespoon of the vegetable oil in a wok or a large pan. Add the tofu and cook for about 5 minutes, stirring, until golden brown all over. Transfer to a plate.
Stir fry spices: Heat the remaining oil. Add the chilies, garlic, ginger, and the white scallion parts to the wok and stir fry for 30 seconds.
Add the sauce and stir until thickened, about 1 minute. Add the tofu and stir to coat. Add the broccoli and mix carefully but thoroughly to coat everything with the sweet-sour sauce.
Serve: Sprinkle the dish with sesame seeds and the green parts of the scallions. Serve immediately.