Set up a steamer pot over medium heat. Cut the kabocha into wedges and steam until very tender, about 15 minutes. Remove from heat and allow to cool.
Once squash is cool enough to handle, scrape the flesh from the skin into a large bowl. Mash with a potato masher or fork until smooth. Set aside. (Kabocha may be prepared up to three days in advance. Store covered in the refrigerator until ready to use).
Preheat oven to 350° F.
Cream butter in large bowl (this may be done by hand with a whisk, but an electric or stand mixer may produce a fluffier texture. Add the oil and mix until fully combined. Follow that with the sugar, again mixing completely.
Mix in the eggs one at a time. Follow with the flour, and mix until just combined and no longer dry. Do not over-mix, or your muffins will end up very dense.
Gently fold in ¾ cup of the mashed kabocha and the chopped curry leaves. Mix just until the batter has fully taken on the orange of the kabocha and little to no white remains.
Spoon batter into lined muffin tins and bake for 18-24 minutes, or until they pass the toothpick test (i.e. it comes out clean). If using the flour we recommend, the muffins will only brown slightly, so this is not the best indicator of how well done they are.
Once the muffins have cooled enough to handle, move them from the tin to a wire rack to cool completely. Store covered to prevent them from drying out.