Arrange tofu in a single layer on a baking sheet lined with several layers of paper towels. Top with more paper towels and another baking sheet; weight with canned goods. Let tofu drain, about 20 minutes.
Meanwhile, cook broccoli (stalks and florets) in a large pot of boiling salted water until bright green, 2 minutes. Drain; set aside.
Heat oil in a large nonstick skillet over medium heat. Add tofu; cook until golden brown, turning gently halfway through, 10 to 15 minutes. Transfer to paper towels.
Meanwhile, make sauce: In a small bowl, whisk together soy sauce, vinegar, red-pepper flakes, garlic, cornstarch, and 3/4 cup water; set aside.
Add broccoli to skillet; cook over high heat, stirring often, until crisp-tender, 2 to 3 minutes. Whisk reserved sauce to combine; pour into pan. Return tofu to pan; stir to coat, about 1 minute more. Serve
topped with cashews.