Wrap tofu block in paper towels and gently press to remove excess water. Cut into 1-inch cubes then place cubes on a large plate lined with paper towels. Place more paper towels on top of the tofu and gently press to dry. Don't press too hard, this makes the tofu tough and dry.
Combine 1/4 cup water, 2 tablespoons hoisin, 2 tablespoons sesame oil, 2 tablespoons soy sauce, and the 1/2 teaspoon pepper in a medium bowl and whisk until smooth.
Heat 2 tablespoons of the vegetable oil in a large nonstick frying pan or wok or cast iron skillet over high heat. Season the oil with salt and pepper. Once shimmering, add tofu and spread out into a single layer. If it doesn't all fit in a single layer, do this step in batches. Cook undisturbed until golden on the bottom. Flip and repeat until brown on all sides, about 7 minutes per batch. Remove from pan and return pan to heat.
Add remaining 2 tablespoons oil to the pan. Once shimmering add 8 ounces mushrooms, give a toss so they are coated in oil then let cook, undisturbed until golden on the bottom, about 2 minutes. Give a quick stir then add the tablespoon of garlic and tablespoon of ginger and stir-fry 1 minute. Add chopped bell pepper and 8 ounces peas and cook 2 minutes more, then add sauce, tofu, and scallions and cook 3 to 4 minutes more, stirring often, until peas are tender.
Serve, garnished with extra thinly sliced scallions over steamed rice or noodles.