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1 large carrot, peeled and thinly sliced
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One 3-inch piece fresh ginger, chopped, plus 1 teaspoon finely grated fresh ginger
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1 cup fresh carrot juice
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2 tablespoons sugar
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1 tablespoon water
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2 tablespoons mustard powder
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1/4 teaspoon Asian sesame oil
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Salt
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In a saucepan, steam the carrot until tender, about 8 minutes. Let cool.
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In a food processor or blender, pulse the chopped ginger until minced. Scrape into a small sieve set over a medium bowl and press to extract the juice; discard the solids. Stir the carrot juice into the ginger juice.
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In a small saucepan, combine the sugar and water and bring to a boil. Remove from the heat and whisk to dissolve the sugar. Let cool. Whisk in the mustard powder, then cover and let stand for 5 minutes. Whisk in some of the carrot-ginger juice to loosen. Transfer the mixture to a blender. Add the remaining carrot-ginger juice and steamed carrot and puree. Transfer the sauce to a bowl. Stir in the grated ginger and sesame oil and season with salt. Transfer the sauce to jars and refrigerate.
Make Ahead
The sauce can be refrigerated for 1 month.