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To prepare dressing:
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Preheat oven to 400F.
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Rub excess papery skin off garlic head without separating cloves.
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Slice the tip off, exposing the ends of the cloves.
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Place the garlic head on a piece of foil, drizzle with 1 tablespoon olive oil and wrap into a package.
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Put in a baking dish and bake until the garlic is very soft, 40 minutes to 1 hour.
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Unwrap and let cool slightly.
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Increase oven temperature to 450 degrees F.
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Squeeze the garlic pulp into a blender or food processor (discard the skins). ]
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Add the remaining 3 tablespoons olive oil, lime juice, vinegar, ginger, sesame oil and soy sauce; blend or process until smooth.
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Season with pepper.
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To prepare salad:
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Toss asparagus with 2 teaspoons oil, 1/4 teaspoon salt and pepper in a large bowl.
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Spread in a single layer on a rimmed baking sheet.
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Roast, stirring once halfway through, until tender and browned, 10 to 15 minutes.
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Meanwhile, season a wooden salad bowl by rubbing with 1/2 clove garlic and a pinch of salt.
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Chop the garlic and add to the bowl along with all the greens.
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(If leaves are large, tear them into bite-size pieces first.)
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Pour 1/4 cup of the dressing over the greens.
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(Cover and refrigerate the remaining 1/2 cup dressing for up to 3 days.)
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Sprinkle the salad with sesame seeds and the asparagus; toss and serve.
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Note: To toast sesame seeds, cook in a small dry skillet over medium-low heat, stirring constantly, until fragrant and lightly browned, 2 to 4 minutes.