Rinse the jasmine rice in a fine-mesh strainer until the water runs mostly clear, 30–60 seconds; combine with 1.5 cups water in a small saucepan, bring to a boil, then cover, reduce to low, and simmer 12 minutes; remove from heat and rest covered 10 minutes, fluff, spread on a tray to cool 20 minutes, then refrigerate until cold.
Beat the egg with the kosher salt until loosened and foamy; set aside.
In a small bowl, stir together the light soy sauce, dark soy sauce, Shaoxing wine, white pepper, and MSG; set the sauce by the stove.
Set a wok or large skillet over high heat until just smoking; add 1 tbsp peanut oil, swirl, pour in the egg mixture, and push into soft curds, 30–45 seconds; slide to a plate.
Return the pan to high heat and add 2 tbsp oil; add shiitake mushrooms (stems removed, sliced 0.25 in thick) and stir-fry until browned and their moisture evaporates, 3–4 minutes.
Make a well, add the scallions (whites first; reserve greens) and the garlic, and stir-fry until fragrant, about 30 seconds.
Add the chilled rice, breaking up clumps; spread into an even layer and let it heat undisturbed 45–60 seconds, then stir-fry until steamy.
Stir the sauce, drizzle it around the sides of the hot pan, and toss to coat; continue stir-frying until the grains separate and look dry-glossy, 2–3 minutes.
Return the scrambled curds, add the reserved greens, and drizzle in the sesame oil; toss to combine and taste, adjusting seasoning if needed; serve immediately.