Rehydrate the dried shiitake mushrooms according to package directions, then quarter them; drain and set aside.
Blanch or briefly cook the trimmed green beans in boiling water until they are bright green and very crisp; drain and set aside.
Heat the vegetable oil in a large saucepan over medium heat.
Add the sliced shallots and green curry paste to the pan and stir until the shallots soften, about 2 minutes.
Pour in the coconut milk, then stir in the fish sauce and brown sugar and bring the mixture to a simmer.
Add the cubed tofu, blanched green beans, rehydrated mushrooms, and red bell pepper strips; stir gently and cook until everything is heated through, about 3–5 minutes.
Stir in the chopped basil and fresh lime juice, then taste and adjust seasoning with salt and pepper as needed.
Serve the curry divided among bowls or over jasmine rice and garnish with basil sprigs and lime wedges.