Rice prep (the day before). Cook the jasmine rice, spread it on a tray or shallow container so it cools fast, and refrigerate within 1 hour of cooking. Chill uncovered or loosely covered overnight so the surface dries. Break up any clumps with your hands before cooking.
Pound the garlic and chiles into a coarse paste with a mortar and pestle (or mince). Stir the oyster sauce, fish sauce, soy sauce, and sugar together in a small bowl.
Heat 1 tbsp oil in a wok or large skillet over high heat. Pour in the beaten eggs and scramble just until set but still soft; remove and set aside.
Add the remaining 1 tbsp oil to the same pan over medium-high heat. Add the garlic-chili paste and onion; stir-fry 1 minute until fragrant and the onion softens.
Add the cold rice. Toss and press with a spatula, breaking up every clump, and stir-fry 2 minutes until the grains are hot and some are lightly toasted.
Pour the sauce over the rice and toss 2 minutes until every grain is coated and the pan is dry-ish — the rice should sizzle, not stew.
Return the eggs to the pan. Turn off the heat, add the basil, and toss until wilted, about 1 minute.
Serve hot with lime wedges and sliced cucumber.