Heat oil and butter in a large pot or high-sided skillet over medium heat. When butter is melted, add onion and shallots and cook until tender and translucent, 6 to 8 minutes.
Add garlic, ginger, and curry powder and cook until fragrant, 1 minute more. Add tomato paste and cook until darkened slightly, 1 to 2 minutes more.
Add coconut milk and water and bring to a simmer. Add chicken and cook, stirring occasionally, until cooked through, 6 to 8 minutes. If the sauce is too thick, add more water, a tablespoon at a time, until the desired consistency is reached.
Stir in lime juice and garnish with mint and cilantro. Serve hot with rice.