If you’ve never cooked tofu before, now is the perfect time to give it a try! Cornstarch helps give the tofu a crispy crust.
Use chef’s knife to cut tofu into ¾-inch cubes. Line rimmed baking sheet with three layers of paper towels. Spread tofu cubes on baking sheet and let drain for 20 minutes.
Slice block of tofu lengthwise (the long way) into four ¾-inch-thick slabs.
Lay slabs flat and cut each slab in half lengthwise to make 2 long fingers (about ¾ inch thick).
Cut tofu fingers crosswise (the short way) into ¾-inch cubes.
While tofu drains, in medium bowl, whisk together broth, soy sauce, vinegar, sugar, 2 teaspoons cornstarch, and sesame oil.
Gently pat tofu dry with more paper towels. Transfer to second medium bowl. Add remaining ⅓ cup cornstarch to bowl. Use rubber spatula to gently stir tofu to coat until there is no dry cornstarch remaining.
In 12-inch nonstick skillet, heat 2 tablespoons vegetable oil over medium heat for 1 minute (oil should be hot but not smoking). Carefully add tofu to pan and use rubber spatula to spread it into even layer. Cook, stirring often, until golden brown, 10 to 12 minutes. Turn off heat. Transfer tofu to large plate lined with paper towels.
Add remaining 1 tablespoon vegetable oil and green beans to skillet and return to medium heat. Cover and cook until green beans are bright green, about 3 minutes. Use oven mitts to remove lid. Continue to cook, stirring occasionally, until green beans are spotty brown all over, about 3 minutes.