In a small bowl, whisk together the fish sauce, light soy sauce, oyster sauce, granulated sugar, and ground white pepper until the sugar dissolves; set aside.
Heat a wok or large skillet over medium-high heat until hot. Add 1 tbsp vegetable oil and the kabocha squash. Stir-fry 3–4 minutes, then add the water, cover, and steam until the squash is just tender and the liquid evaporates, 2–3 minutes. Transfer the squash to a plate.
Return the wok to medium-high heat and add the remaining 2 tbsp vegetable oil. Add the yellow onion and stir-fry until translucent and lightly golden at the edges, 2–3 minutes. Add the garlic and cook until fragrant, about 30 seconds.
Push the aromatics to one side. Pour in the beaten eggs and scramble just to soft curds, 30–60 seconds.
Add the cooked, chilled jasmine rice. Stir-fry, breaking up clumps, until the grains are hot and separate, 2–3 minutes.
Return the cooked squash to the wok, pour in the sauce mixture, and add the scallions. Stir-fry until the rice is evenly coated, glossy, and steaming, 1–2 minutes. Taste and adjust seasoning if needed.
Serve immediately with cucumber slices, torn Thai basil leaves, and lime wedges for squeezing at the table.