Heat the oil in a large pan/skillet on high heat. Sauté the onion and carrot until the onion becomes soft and transparent. Add the chicken and fry until just beginning to brown. Add the garlic and curry paste (if using), stirring together quickly to blend in the heat of the pan for about a minute until fragrant.
Add the butternut pieces and green beans; keep cooking while stirring occasionally for 5 minutes. Stir the coconut milk through the ingredients and bring to a simmer. Reduce heat to low; add in the sugar and salt to your taste (I use about 1 teaspoon), and continue to simmer for about 10-15 minutes, or until the pumpkin pieces have softened. (If you prefer a creamier pumpkin curry sauce, simmer for 5-10 minutes longer until the pumpkin has completely melted through the sauce. YUM.
Serve over steamed rice or cauliflower rice.