How to Make It
Preheat the oven to 400°. Line a rimmed baking sheet with foil. Place the squash halves cut side up on the prepared baking sheet, drizzle with the sesame oil and season generously with salt and pepper. Turn the squash halves over and bake for about 1 hour, until the squash is easily pierced with a fork. Remove from the oven and let cool slightly, about 20 minutes.
Using a fork, scrape out the squash flesh. Transfer to a colander set over a large bowl, toss with 2 teaspoons of salt and let drain at room temperature for 15 minutes. Pat dry with paper towels.
In a small bowl, stir the fish sauce with the lime juice and brown sugar.
In a large nonstick skillet, heat 2 tablespoons of the vegetable oil until shimmering. Add the garlic and chiles and cook over moderately high heat, stirring, until softened, about 1 minute. Add the carrots, bok choy, bell pepper and a generous pinch of salt and cook over moderately high heat, stirring often, until the vegetables are softened, about 4 minutes. Add 1/4 cup of the sauce and cook for 1 minute. Remove the skillet from the heat and stir in 1 cup of the scallions and 1/4 cup of the cilantro. Transfer the vegetable mixture to a large bowl. Wipe out the skillet.
In the same skillet, heat 1 tablespoon of the vegetable oil until shimmering. Add the eggs and cook over moderate heat, swirling the pan to coat the bottom but not stirring, until the bottom is just set, about 1 minute. Using a thin spatula, carefully flip the omelet and cook until the bottom just sets, about 15 seconds. Carefully slide the omelet onto a work surface. Gently roll it into a log and slice it crosswise into 1/4-inch-thick strips.
In the same skillet, heat the remaining 2 tablespoons of vegetable oil until shimmering. Add the drained spaghetti squash and cook over moderately high heat, stirring occasionally, until beginning to brown, about 5 minutes. Add the remaining 1/4 cup of sauce, stir to combine and cook for 1 minute. Using a slotted spoon, return the vegetables to the skillet along with the omelet strips and cook, stirring occasionally, until heated through, about 2 minutes. Remove the skillet from the heat, stir in 1/4 cup of the peanuts and the remaining 1/4 cup each of scallions and cilantro. Transfer the pad Thai to a platter and garnish with cilantro and the remaining 1/4 cup of peanuts. Serve with lime wedges.