In a small bowl, whisk together 1/4 cup soy sauce, 3 tablespoons packed dark brown sugar, 1 1/2 tablespoons fresh lime juice, 1 tablespoon water, 1 tablespoon Sriracha sauce, and 2 cloves minced garlic until the sugar is mostly dissolved. Set the sauce aside.
Heat 1 tablespoon vegetable oil in a large nonstick pan over medium-high heat until the oil is shimmering.
Sprinkle the 12 ounces extra-firm tofu (patted dry and cut into 3/4-inch cubes) with salt. Add the tofu to the hot pan and cook, stirring or turning occasionally, until browned on most sides, about 5 minutes. Transfer the browned tofu to a medium bowl and set aside.
Add another 1 tablespoon vegetable oil to the same pan and return to medium-high heat. Add the 1/2 small thinly sliced yellow onion, the white parts of the 4 green onions, and 1 cup shredded carrots. Cook, stirring occasionally, until the vegetables are crisp-tender, about 3 minutes. Transfer these vegetables to the bowl with the tofu.
Add the remaining 1 tablespoon vegetable oil to the pan and heat over medium-high. Add 3 cups thinly sliced green cabbage and cook, stirring frequently, until the cabbage has wilted, 3 to 5 minutes.
Add the green parts of the 4 green onions, 1 pound cooked Asian-style noodles, and the tofu–carrot–onion mixture back to the pan with the cabbage. Use tongs or a spatula to separate and distribute the noodles.
Pour the reserved sauce evenly over the contents of the pan. Stir well and cook for 1 to 2 minutes more, until everything is heated through and the sauce coats the noodles and vegetables.
Remove the pan from the heat. Serve the Sriracha noodles with tofu immediately, with lime wedges and extra Sriracha sauce on the side.