How to Make It
In a large nonreactive skillet, heat 1 tablespoon of the peanut oil over high heat. Add the mushrooms in an even layer and cook, turning once, until browned, about 3 minutes on each side. Add 1 tablespoon of the soy sauce, toss to coat and cook for 2 minutes longer. Transfer the mushrooms to a plate.
Reduce the heat to low and add the remaining 1/2 tablespoon peanut oil to the skillet. Add the garlic slices in an even layer and cook until golden brown, about 3 minutes. Increase the heat to moderate, add the onion and cook, stirring, until wilted, about 3 minutes.
Add the white wine and scrape the bottom of the pan to loosen the browned bits. Add the oyster sauce and the remaining 1 tablespoon soy sauce and bring to a simmer.
Meanwhile, dissolve the cornstarch in 1 cup of water and stir it into the sauce in the skillet. Return the mushrooms to the skillet and simmer the sauce until thickened, about 4 minutes. Stir in the sesame oil and season with pepper. Remove from the heat.
In a medium saucepan, heat the vegetable oil over moderately high heat until it registers 360° on a deep-fry thermometer. Gently pat the tofu dry with paper towels and carefully lower 4 slices into the hot oil. Stand back—the oil may spatter. Fry the tofu until golden brown, gently scraping it off the bottom of the pan with a slotted spatula if it seems to be sticking, about 4 minutes. Using the spatula, transfer the tofu to a baking sheet. Pat it with paper towels to remove excess oil, and keep warm in a low oven. Repeat the procedure with the remaining tofu slices.
Rewarm the mushroom gravy over moderately high heat, stirring. Pour the gravy onto a platter, arrange the tofu slices on top and serve at once.