Heat 2 teaspoons oil in a large nonstick skillet over medium-high heat. Add tofu and cook, stirring every 2 or 3 minutes, until browned, 6 to 8 minutes total. Transfer to a plate.
Heat the remaining 2 teaspoons oil over medium-high heat. Add sweet potato and cook, stirring occasionally, until browned, 4 to 5 minutes. Add coconut milk, broth and curry paste to taste. Bring to a boil; reduce to a simmer and cook, covered, stirring occasionally, until the sweet potato is just tender, about 4 minutes. Add the tofu, green beans and brown sugar; return to a simmer and cook, covered, stirring occasionally, until the green beans are tender-crisp, 2 to 4 minutes. Stir in lime juice and salt. Sprinkle with cilantro and serve with lime wedges.