1. In a saucepan, steam the carrot until tender, about 8 minutes. Let cool.
2. 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.
3. 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.