Place yam into a large pot and cover with water; bring to a boil. Reduce heat to medium-low and simmer until tender, about 15 minutes. Drain.

Heat 1 teaspoon peanut oil in a wok or large skillet over medium heat; cook and stir green bell pepper and onion until tender, 2 to 3 minutes. Transfer to a bowl using a slotted spoon. Mix remaining 2 teaspoons peanut oil and green curry paste in the wok; stir for 30 seconds.

Increase heat to medium-high; add chicken to the wok and cook until golden, 3 to 5 minutes. Reduce heat to medium-low. Pour in coconut milk, chicken broth, and fish sauce; bring to a gentle simmer. Stir in yam, green bell pepper, and onion; simmer for 5 minutes.

Stir spinach and basil gently into the wok. Simmer, covered, until flavors combine, 4 to 5 minutes more.