Heat half the oil in a large, non-stick skillet set over medium heat. Whisk eggs with red pepper flakes and salt. Pour into the skillet and cook, covered, for 1 to 2 min. or until firmly set. Using a spatula to lift the edge, roll up the egg and remove from pan. Cut into thin strips and reserve.
Add the remaining oil, onion and carrot to pan. Cook, stirring often, for 5 min. Increase the heat to medium-high and cook, stirring, for 3 min. or until the onion is golden. Add cooked rice, cucumber, peanut sauce and lemon or lime juice. Cook, tossing, until heated through. Garnish with reserved eggs.