In a small bowl combine the oyster sauce, soy sauce, and sugar. In a large nonstick skillet, heat 1 tsp. sesame oil over medium-high heat. Add the eggs, and cook until softly scrambled. Transfer to a plate; gently break into small pieces. Wipe the pan clean.
Add 1 tsp. sesame oil and the broccolini to the pan; cook 1 minute. Add 1/3 cup water, cover and steam until just crisp-tender, about 2 minutes. Transfer the broccolini to a plate. Add 2 Tbs. olive oil and onion to the pan; cook 1 minute. Add the ginger and garlic; cook 30 seconds. Add the rice and carrots; cook and stir 2 minutes.
Stir in the broccolini, egg, and sauce. Cook and stir 2 minutes more.
Prepare 1 2/3 cups jasmine rice according to package directions. Cool, and store in the fridge.