Remove the tofu from its package and wrap the block in an absorbent kitchen towel or several layers of paper towel; place on a plate and weigh down with a heavy pan or other sturdy item for at least 10 minutes to press out excess water.
Preheat the oven to 425°F (220°C). After pressing, cut the tofu into 1-inch cubes.
Heat 2 tablespoons of the oil in a cast iron or ovenproof skillet over medium-high heat. Add the tofu and 1/2 teaspoon salt, and cook, turning occasionally, until all sides are golden, about 5–7 minutes.
Transfer the skillet to the preheated oven and bake the tofu for 15 minutes, until crispy on the edges; remove and set aside.
While the tofu bakes, bring about 3 cups of water to a boil in a pot and blanch the broccoli florets for 1 minute; drain and set aside.
In a small bowl, whisk together the vinegar, 1/4 cup water with 1 teaspoon vegetable bouillon paste, soy sauce, fermented black bean sauce, sugar, toasted sesame oil, and arrowroot powder until smooth.
Wipe the skillet if needed, then warm the remaining 1 tablespoon oil over medium heat. Add the garlic, ginger, and chopped green onions and sauté until fragrant, about 30–60 seconds.
Add the blanched broccoli to the skillet and cook, stirring, for about 5 minutes until heated through.
Give the sauce a quick stir, pour it into the skillet, and cook, stirring, until the sauce thickens and coats the vegetables.
Return the crispy tofu to the skillet and gently toss to coat with the sauce. Serve immediately over cooked brown rice.