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First, make the pickles:
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Slice the cucumbers into 1/8-inch slices, and places slices in a bowl. Add the sugar and salt, and toss to coat the slices evenly. Transfer them to a colander, and allow to drain and soften for 20 to 30 minutes. Rinse well, then drain, place in a bowl, and add the vinegar. Toss to coat, then taste and adjust vinegar as needed. Transfer pickles to a storage container, and refrigerate for up to a week in advance.
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Next the Bibimbap:
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Heat oven to 400 degrees. Peel the squash and cut it crosswise, separating the long neck from the bulbous bottom part. Slice the neck into 1/2-inch-thick dominoes; scoop out seeds from the bottom and slice the squash 1/2 inch thick.
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Whisk together 2 tablespoons neutral oil, the soy sauce, brown sugar or palm sugar and gochujang or sambal oelek or Sriracha.
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Place squash and mushrooms on one baking sheet, but do not mix them together. Place broccoli rabe on another baking sheet. Divide sauce between the two pans and use your hands to toss the vegetables so they're evenly coated.
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Transfer both pans to the oven. Cook broccoli rabe for 5 to 8 minutes, until collapsed and the thicker parts of the stems are tender. Cook mushrooms for 15 to 20 minutes, until juicy and slightly shrunken and remove from baking sheet. Return squash to oven and cook 5 to 15 minutes longer, until caramelized and tender. Cover the vegetables with foil until ready to serve.
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If you'd like, make crispy-base bibimbap rice: Just before serving, heat 1 tablespoon neutral oil and the sesame oil in a wide skillet over medium heat. Press rice into the skillet, making a thick cake. Let cook without disturbing for 4 to 5 minutes, until a golden brown crust forms on the bottom of the rice. (If you skip this step, use freshly cooked rice instead.).
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While the crispy rice is cooking, fry the eggs: Heat a large skillet over medium-high heat. Add enough neutral oil to liberally coat pan. Crack in 2 eggs and sprinkle with salt. Tilt the pan so some of the oil runs over the edges of the egg whites, lower heat to medium-low and cook 1 minute. Sprinkle with about 1/4 teaspoon water (or soy sauce), cover and cook another minute, until whites are set. Carefully remove to a plate. Repeat with remaining eggs.
