Remove tofu from package, wrap in a clean kitchen towel or paper towels, and press under a heavy skillet or books for at least 10 minutes to remove excess moisture.

In a small bowl whisk together soy sauce, vegetable broth, rice vinegar, brown sugar, and sesame oil; set aside. In a separate small bowl dissolve 1 teaspoon cornstarch in 2 teaspoons water and set the slurry aside.

Heat 2 tablespoons oil in a 12-inch skillet over medium-high heat until shimmering. Add tofu in a single layer without crowding and cook undisturbed 3–4 minutes per side until deeply golden; you may need two batches. Remove browned tofu to a plate and keep warm.

Add the remaining 1 tablespoon oil to the hot skillet. Add mushrooms in a single layer and let sear 2–3 minutes without stirring, then stir and add bell pepper and a pinch of salt. Cook 1–2 minutes until vegetables begin to brown. Push vegetables to the side and add garlic and ginger; cook 20–30 seconds until fragrant.

Return tofu to the pan, stir the sauce, and pour over the tofu and vegetables. Bring to a simmer, then slowly stir in the cornstarch slurry. Cook 30–60 seconds until the sauce thickens and becomes glossy. Remove from heat and sprinkle with sliced green onions and a squeeze of lime if using.

Serve immediately over steamed rice or noodles.