Combine the rice with 2 cups water in a medium saucepan over medium heat (or follow the ratio and cooking instructions on your rice package). Season with salt and add one of the pieces of ginger. Bring to a boil, cover and reduce heat to low. Cook for 20 minutes, then remove from the heat and let rest.
Meanwhile, slice the mushrooms to yield 1 cup. Chop the onion. Mince the garlic to yield 1 tablespoon. Peel and grate the remaining piece of ginger to yield 1 tablespoon. Slice the carrots crosswise. Reserve.
Preheat a large skillet over medium-high heat. Add the oil and swirl to coat. Add the mushrooms and cook, stirring occasionally, until browned, about 3 minutes. Push the mushrooms to the edges of the skillet and add the onions. Season the onions with salt to draw out some of their moisture and cook, stirring. When the onions start to turn golden, add the chile, garlic, ginger and carrots. Cook, stirring, for 1 minute, then add the curry powder. Stir to combine, then pour in the shellfish broth and simmer for 3 minutes.
Next, stir in the coconut milk and simmer until the curry starts to thicken and coats the back of a spoon, about 2 minutes. Add the shrimp to the skillet in a single layer. Reduce the heat to low and let simmer for 6 minutes. (The slower the shrimp cook, the more tender they will be.)
Meanwhile, remove enough of the cilantro and basil leaves from their stems to yield 1/2 cup each; discard the stems.
Flip the shrimp and cook until just cooked through, about 6 minutes. Add the fish sauce and lime juice and season with salt. Stir in the cilantro and basil leaves.
Place the rice in individual bowls, spoon the shrimp curry on top and serve with the lime wedges.