Heat a Dutch oven or similar large saucepan over medium heat. Once hot, add the oil, red pepper, and onion. Cook, stirring often, until the onion is turning translucent, 3 to 4 minutes.
Stir in the garlic and ginger. Then, stir in the curry paste. Cook for 1 minute, stirring constantly. The pot should be very fragrant.
Add the coconut milk and broth. Stir until smoothly incorporated. Bring to a steady simmer. Let simmer to reduce for 10 minutes (if you haven't cubed your pumpkin/squash yet, this is a great time to do it).
Add the cubed pumpkin to the soup and let simmer, stirring every now and then to make sure it cooks evenly and doesn't stick to the bottom), until the pumpkin is tender (be sure to taste a few pieces), about 10 minutes.
While the squash simmers, cut the broccoli into florets.
Add the broccoli to the curry, stirring to combine (some of the pumpkin cubes may start breaking down a little, which is fine and delicious). Simmer 5 additional minutes, until the broccoli is bright green and tender.
Stir in the lime juice, soy sauce, and fish sauce. Serve warm with rice and a sprinkle of cilantro, plus additional soy sauce as desired.
Notes
TO CUBE A PUMPKIN: Using a sharp knife, carefully cut out the pumpkin stem, just as you would a carving pumpkin. Scoop out the seeds (discard or make Roasted Pumpkin Seeds). Slice the pumpkin in half lengthwise, then peel it with a vegetable peeler. Slice into quarters, then strips, and cube.
OPTIONAL: For an easy protein boost, stir 1 (15-ounce) can of rinsed and drained chickpeas into the curry with the broccoli and bell pepper.
TO STORE: Keep leftover pumpkin curry in an airtight container in the refrigerator for 3 to 4 days
TO REHEAT: Warm in a pan over medium heat or reheat the curry in the microwave.
TO FREEZE: I don’t recommend freezing curries made with coconut milk.