Heat the oil in a large non-stick skillet or wok over a high flame. Add the shallots, carrots, and radishes and sauté until the shallots are soft and the radishes are vibrantly pink, about 3 minutes. Add the garlic and cook until fragrant, about 30 seconds more. Add the rice and stir fry, stirring constantly, until well coated in the shallot mixture and beginning to brown, about 3 minutes. Stir in half the basil, fish sauce, and juice of half the lime. Cook for another minute. Remove from heat.
Divide the rice among two plates and garnish with the remaining basil. Cut the remaining lime into wedges and serve along with the rice.