In a large, deep skillet or dutch oven (that has a lid) over medium heat, saute onion until lightly browned. About 3-4 minutes.
Add eggplant and kabocha to the pan, and then add coconut milk, water, curry paste and soy sauce. Bring to a simmer, then reduce heat to low and cover. Simmer until the kabocha can be pierced with a fork but still has a little resistence in the center. About 7-10 minutes.
Add broccoli, green beans and bell pepper to the pan and stir to incorporate. Return the lid and simmer until the vegetables are tender stirring occasionally. About 5-7 minutes longer.
Add the lime juice and stir to distribute.
Serve with rice or quinoa and a sprinkle of cilantro if you wish.