Heat oil in a big pot or dutch oven over medium heat.
Add the onions and saute for about 3-5 minutes or until soft and slightly translucent. Stir occasionally to make sure they don't burn or stick to the pan.
Add lemongrass, ginger, curry paste, garlic, peppers, and kaffir lime leaves; saute for another 2 minutes, stirring frequently.
Add broth and coconut milk. Give a good stir to make sure nothing is stuck to the bottom of your pot and to incorporate everything well.
Add the sweet potatoes and stir to incorporate them fully.
Turn up the heat and bring to a full boil, then turn the heat down to low and simmer covered for 15-20 minutes, or until your sweet potatoes are cooked through and soft.
Add the kale and cook for another 3-5 minutes, so all the kale is cooked and soft.
Take off the heat, and add lime juice. Give it a good stir and adjust seasoning to taste.
Serve with suggested toppings. Once dished, top with another squirt of lime and chopped fresh basil (if using).