Scoop thick coconut cream from top half of both cans (reserving watery milk in cans for later), and add to a medium Saucepan. Heat Saucepan over medium. Stir to mix and bring to a boil, then boil for 1 minute without stirring.

Stir in curry paste and mix until combined and smooth (if you are sensitive to heat, you can reduce the amount of curry added). Simmer without stirring until fragrant and coconut cream starts to release some oil, 3–5 minutes.

Add chicken and cook, uncovered, stirring occasionally, until it is halfway done, about 5 minutes. Add remaining coconut milk, increase heat to medium-high, and bring to a boil.  Add sweet potatoes and chicken broth, bring back to a boil, and simmer until chicken is cooked through and potatoes are fork-tender, 5–7 minutes. Add fish sauce and sugar. Taste and adjust seasoning as needed. Add basil and lime leaves and stir to  combine. Serve with rice.