In a medium bowl, toss the cubed tofu with soy sauce and cornstarch until evenly coated. Let it marinate for about 10 minutes.
Heat 1 tablespoon of vegetable oil in a large skillet or wok over medium-high heat. Add the marinated tofu and cook until golden brown on all sides, about 5-7 minutes. Remove the tofu from the skillet and set aside.
In the same skillet, add the remaining tablespoon of vegetable oil. Add the sliced mushrooms and bell pepper, and stir-fry for about 3-4 minutes until they start to soften.
Add the broccoli, garlic, and ginger to the skillet. Stir-fry for another 3-4 minutes until the broccoli is bright green and tender-crisp.
Return the cooked tofu to the skillet and drizzle with sesame oil. Toss everything together and season with salt and pepper to taste. Cook for an additional 1-2 minutes to heat through.
Serve the stir-fry over cooked rice or noodles, garnished with chopped green onions.