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1
Trim off and discard the tough ends of the broccoli.
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2
Cut off the heads and separate the heads into roughly 2-inch pieces.
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3
Peel the broccoli stems with a vegetable peeler or a sharp knife, discard the peelings, and slice the stems lengthwise into 1/2-inch-thick slabs.
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4
Slice each slab lengthwise into 1/2-inch-wide batonettes.
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5
Combine the onion, garlic, chile, and 1/4 cup of the olive oil in a large saute pan over medium heat; season with the salt and pepper and saute until the onion begins to soften, 4 to 5 minutes.
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6
Add the broccoli stems and saute for 2 to 3 minutes to soften them slightly.
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7
Add the broccoli heads and 3/4 cup of the remaining olive oil and cook for 2 to 3 minutes, stirring to prevent the vegetables from browning and so they wilt evenly.
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8
Reduce the heat to low, and add the remaining 1/2 cup of olive oil.
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9
Cook, gently stirring only as necessary to prevent the broccoli from sticking to the pan but not so much or so vigorously that the broccoli heads fall apart, until the broccoli is very tender but still holding its shape, 45 minutes to 1 hour.
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10
Taste for seasoning and add more salt, if desired.
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11
Carefully lift the broccoli out of the oil and transfer it to a platter to serve.
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12
Reserve the oil to make the dish a second time or use it to saute or drizzle over vegetables.
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13
Serve warm or at room temperature.
