1. Carefully spoon off about 1 cup of the top layer of cream from one can of coconut milk -- this layer will be thick and possibly solid. Place the coconut cream and curry paste in a large Dutch oven and bring to a simmer over high heat, whisking to blend, about 2 minutes. Maintain this brisk simmer and whisk frequently until almost all of the liquid evaporates, 3 to 5 minutes. Reduce the heat to medium-high and whisk constantly until the cream separates into a puddle of colored oil and coconut solids, 3 to 8 minutes. (You should hear the curry paste starting to fry in the oil.) Continue cooking until the curry paste is very aromatic, 1 to 2 minutes.
2. Whisk in the remaining coconut milk, the fish sauce, and the brown sugar. Bring back to a brisk simmer and cook until the flavors meld and the sauce thickens, about 5 minutes. Season the chicken with salt and add it to the pot, stirring until the pieces are separated and evenly coated with the sauce, about 1 minute. Stir in the broccoli and mushrooms and bring back to a brisk simmer over medium heat. Cook until the vegetables are almost tender, about 5 minutes. Stir in the bell pepper and fresh chile, if using, and cook until the vegetables are crisp-tender, about 2 minutes. Off the heat, stir in the lime juice, basil, and mint. Serve immediately.