Shake the can of coconut milk or stir it well (this creates a consistent thickness, since the fat often solidifies at the top of the can).
In a 3- to 4-quart saucepan or wok over medium heat, simmer 1/2 cup of the coconut milk, stirring occasionally, until reduced by about half, 3 to 5 minutes. It will get very thick and shiny and may or may not separate; either is fine.
Add the curry paste, whisk well, and cook, continuing to whisk, for 1 minute. Whisk in the broth, sugar, fish sauce, and remaining coconut milk. Bring to a simmer over medium-high heat.
Add the carrots, ginger slices, lemongrass pieces, and whole lime leaves and continue to simmer, adjusting the heat as necessary. After 2 minutes, add the shrimp, bell peppers, button mushrooms, and cabbage and continue to simmer until everything is tender and cooked through, about 3 minutes more.
Remove the curry from the heat. Season to taste with more sugar and fish sauce, and add the basil and cilantro. Transfer to a serving bowl (or serve right out of the pot). Remove the ginger slices, lemongrass pieces, and whole lime leaves or tell your guests to eat around them. Garnish with the basil, lime wedges, red chiles, coconut milk, and cilantro.
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