Preheat oven to 375ºF. Line a rimmed baking sheet with parchment paper, a silicone mat or foil.
Remove tofu from package and pour away water. Slice the tofu into 1-inch cubes.
Line a cutting board with a clean tea towel or 3 layers of paper towels, then arrange the tofu on top in a single layer. Fold the towel over the tofu (or add 3 more paper towels on top), and place something heavy on top to press the excess water out (like another cutting board, weighed down with a heavy pan). Let the tofu drain for at least 20 minutes, up to 3 hours.
Transfer the pressed tofu to a medium mixing bowl and drizzle with the vegetable oil, sesame oil and soy sauce. Gently toss to combine. Sprinkle the cornstarch, panko and sesame seeds over the tofu, and gently toss until evenly coated.
Spread tofu out onto the lined baking sheet in a single layer, leaving a little space between cubes. Bake for 20 minutes, then turn and cooking until the outside of the tofu is golden brown and the pieces look slightly puffed, 10 - 20 minutes more. The longer you bake the tofu, the chewier it will be.
If using immediately, serve while still warm. If saving the tofu for later, let it cool on the baking sheet, then transfer to a refrigerator container. The tofu will keep refrigerated for up to a week.