Wrap tofu in a clean kitchen towel and place in a shallow baking dish or on a rimmed baking sheet. Weigh down with a heavy object. Let sit at least 15 minutes and up to 1 hour.
Meanwhile, coarsely crack peppercorns in a spice mill or with a mortar and pestle; set aside.
Finely grate garlic and ginger into a small bowl; set aside.
Unwrap tofu and cut into 1" cubes. Transfer tofu to a medium bowl. Add cornstarch and salt and toss gently to coat tofu.
Heat oil in a large nonstick skillet over medium-high. Arrange tofu in skillet in a single layer and cook, turning occasionally, until golden brown and crisp all over, 6–8 minutes. Using tongs or a slotted spoon, transfer to a large plate or baking sheet.
Reduce heat to medium and add reserved cracked pepper to skillet. Cook, stirring often, until very fragrant, about 2 minutes, then add asparagus and cook, stirring often, until bright green, about 1 minute.
Add reserved garlic and ginger and cook, stirring constantly, until fragrant, about 1 minute.
Return tofu to pan and gently toss to combine. Increase heat to medium-high, add soy sauce and sugar, and cook, tossing occasionally, until sauce thickens slightly, about 3 minutes.
Remove from heat and add vinegar. Taste and add more salt and/or vinegar if needed.
To serve, divide rice among plates and top with tofu mixture.