Heat a Dutch oven, large saucepan, or deep skillet over medium. Once hot, add the olive oil, garlic, ginger, and onion. Cook, stirring frequently, for 3 to 4 minutes or until the onion is translucent.
Add the curry paste and stir to coat. Cook for 1 minute, until fragrant.
Add the coconut milk. Stir until the sauce is smoothly incorporated.
Add the sweet potato, kale, and chickpeas. Increase the heat to medium high and bring to a simmer.
Let simmer for 5 minutes, until the sweet potato is somewhat tender but still has a bit of chew, then add the broccoli. Simmer 3 to 5 additional minutes, until the broccoli is bright green and tender.
Stir in the lime juice, fish sauce, and coconut sugar.
Remove from heat and let cool for 3 minutes (this will ensure the yogurt does not curdle and gives the flavors a little extra time to marry). Stir in the Greek yogurt. Serve warm with rice and a sprinkle of fresh basil.
