At least 30 minutes (and as many as two or three hours!) before you want to start cooking, press the tofu: Place a dish cloth or paper towels on a cutting board, then the tofu, then another layer of towels. On top of the tofu, place a board and some heavy objects to weight it down. Heat the oven to 400° F and line two sheets with parchment paper.
When the tofu is ready, cut it into 1/2-inch cubes and place in a bowl. In another bowl, place the sliced mushrooms. To each bowl, add 2 tablespoons vegetable or peanut oil, 2 teaspoons sesame oil, 2 teaspoons soy sauce, and a pinch of salt.
To the bowl of tofu, add cornstarch, panko, and sesame seeds and stir to coat.
Spread the tofu out on one baking sheet and the mushrooms out on another. Bake for 30 to 40 minutes, flipping halfway through. When the tofu and mushrooms are finished, mix them together and drizzle with fish sauce, if you'd like, and remaining 1 teaspoon soy sauce. Sprinkle with chives and additional sesame seeds.
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