Pour the oil into a large pot over medium-high heat. Add the lemongrass and let it sizzle in the pot for 1 to 2 minutes. Next, add the shallots and sauté a minute more. Add the ginger and sauté for another 30 seconds to a minute.
Transfer the rice to the pot. Then, add the chicken broth and water. Cover the pot with a lid and bring to boil.
Reduce the heat to low. I usually move the pot to the smallest burner on the stove so that the congee can cook very slowly. Let the congee simmer for 30 minutes. Add the kabocha purée and stir until the purée is well incorporated.
Cover the pot and continue cooking the congee for another 15 minutes. Because of the sugar content in the kabocha, the jook will start to scald at the bottom of the pot. Give the jook a stir, carefully scraping the bottom for any kabocha build-up. Cover the pot again and cook for another 10 minutes.
Turn off the heat and give the congee a stir. The congee should be quite thick at this stage. If you want the congee to be thicker, let the congee cool in the pot, uncovered, for another 15 to 20 minutes. The congee thickens as it cools and the flavor of the kabocha develops further.
Use tongs to fish out the sections of lemongrass. Add the scallions and fish sauce into the pot and stir to incorporate.
Serve the kabocha congee in bowls with Chinese donuts, additional scallions, and fried shallots, if you like. Refrigerate any leftovers in an airtight container for up to 4 days. I usually reheat the congee in the microwave for 1 to 2 minutes. You can also reheat the congee in a saucepan.