Heat 2 teaspoons of the oil in a large nonstick skillet (I use my nonstick wok) over medium-high heat. Add the tofu and cook, stirring every 2-3 minutes, until browned, about 6-8 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, over rice if desired.