Cook the noodles according to package directions. Drain and keep warm.
Meanwhile, heat the oil in a 12-inch skillet over medium-high heat. Add the mushrooms, carrots, and bell pepper and cook, stirring often, until just crisp-tender, about 3 minutes. Add the garlic and ginger and cook, stirring constantly until fragrant, 30 seconds.
Add the coconut milk, broth, curry paste, fish sauce, and brown sugar and cook, stirring often, until mixture comes to a boil. Stir in the chicken and cook just until heated through, about 2 minutes. Stir in the noodles and basil.
Divide evenly among 4 bowls and serve with lime wedges and desired toppings.