Heat 1 tablespoon oil in a large saute pan or wok over high heat. Add ground chicken and cook until browned, about 4-5 minutes. Season with salt and pepper and transfer to a plate.
Heat another tablespoon oil in the same pan. Add chopped bell peppers and cook until softened, about 3 minutes. Add shallots, garlic, Thai chilies, and chopped white scallion ends. Stir to combine and saute for 2 minutes or until softened. Add red curry paste and saute for another minute, stirring to combine.
Add back cooked ground chicken along with rice. Break up rice with a wooden spoon and mix until well combined. Continue to cook until rice is hot.
Season fried rice with fish sauce, soy sauce, sugar, and black pepper. Remove from heat and stir in chopped Thai basil and chopped green scallion ends.
Portion Thai red curry fried rice onto 4 plates and top with fried eggs. Serve with lime wedges if desired.