In a 12-inch frying pan or 14-inch wok over medium heat, stir onion, ginger, and garlic in oil until onion is barely limp, about 2 minutes.
Stir in paprika, curry paste, and rice, mixing well to coat rice. Pour in sherry and cook, stirring to scrape up any browned bits, until sherry is absorbed, about 2 minutes.
Pour in 1 cup chicken broth and stir until broth is mostly absorbed, 3 to 4 minutes. Add mushrooms and coconut milk; stir gently until liquid is mostly absorbed, about 1 minute. Add 2 more cups broth, 1 cup at a time, stirring after each addition until almost absorbed, about 15 minutes total. Rice should be almost tender but still slightly firm to bite. Remove from heat and let stand, uncovered, up to 4 hours.
Return pan to medium-high heat, add 1 more cup broth, and stir until liquid is absorbed and rice is creamy and tender to bite, about 9 minutes. If risotto is too dry or rice isn't quite tender, add a little more broth and cook until risotto reaches desired consistency. Stir in peas, cilantro, and salt to taste. Spoon risotto into serving bowls and garnish each serving with a lime wedge.