How to Make It
Preheat the oven to 400°. Line a rimmed baking sheet with parchment. Place the squash halves cut side up on the prepared baking sheet. Season generously with salt and pepper, then drizzle with 1 tablespoon of oil. Roast, cut side down, for about 1 hour, until easily pierced with a knife. Remove from the oven and, using tongs, flip the squash halves. Let stand at room temperature until cool enough to handle, about 20 minutes.
Use a fork to scrape the spaghetti squash into a colander, separating the strands. (You should have about 4 cups.) Let drain for 20 minutes. Using a spatula, press the squash to remove excess water. Transfer to a large bowl. Add the butter and the 2 tablespoons of chives and toss; season with salt and pepper.
Heat a large cast-iron skillet over high heat until nearly smoking. Add the remaining 2 teaspoons of oil and tilt the pan carefully so the oil coats the bottom. Spoon the squash into the skillet and, using a metal spatula, pat into a large cake. Cook for 2 minutes, without stirring, until the bottom is beginning to brown. Flip the hash browns in several large pieces and cook until browned on the bottom, 2 minutes longer.
Transfer the squash hash browns to plates. Sprinkle with more chives and season with salt and pepper. Serve with fried eggs and a green salad.