To make the sauce, stir ¾ cup of water together with the soy sauce, vegan oyster sauce, sesame oil, and black pepper in a small bowl or cup. Set this aside.
In a separate small bowl or cup, stir together the remaining ¼ cup of cold water and cornstarch. Set this aside.
Coat the bottom of a large nonstick skillet with a tablespoon of the oil and place it over medium heat. Give the oil a minute to heat up, then add the tofu cubes. Arrange them in a single layer and cook them for about 10 minutes, flipping them once or twice until they're golden brown. Remove the tofu from the skillet and transfer it to a plate.
Add the remaining 2 tablespoons of oil to the skillet, then give it a minute to heat up. Add the mushrooms. Spread them out and let them cook undisturbed for two to three minutes, then stir them and let them cook a few more minutes. Repeat one or two more times, cooking the mushrooms until they're tender and browned.
Add the shallots to the skillet, then cook them with the mushrooms for about 5 minutes, stirring occasionally, until they begin to soften.
Push the mushrooms and shallots to the side, then add the garlic, ginger, and the white parts of your scallions. Sauté the mixture until it becomes very fragrant, about 1 minute, stirring constantly.
Return the tofu to the skillet. Stir everything up and add the soy sauce mixture to the skillet. Raise the heat and bring the liquid to a boil, then lower the heat and let it simmer for about 5 minutes, until reduced slightly.
Stir in the cornstarch mixture. Raise the heat and bring the sauce back to a simmer. The sauce should thicken once it begins to simmer. Remove the skillet from heat once it does.
Taste-test the sauce and adjust any seasonings to suit your taste, such as adding more soy sauce, oyster sauce, or black pepper. Stir in the green parts of your scallions.
Serve with rice.