Feel free to scale the recipe down, but after 4 weeks of waiting you might want to have a sizable supply of sauce to reward your efforts and patience. For a vegan version, omit the anchovies and double the amount of dried shiitakes.
Combine in a large saucepan:
Bring to a boil, then reduce the heat and simmer gently for 30 minutes. Meanwhile, warm over medium heat in a skillet, stirring frequently:
The sugar will melt and turn deep brown, about 7 minutes. Slowly and carefully pour the caramelized sugar into the simmering mixture (it will sizzle and spit, so back up a little). Remove from the heat and let cool to room temperature. Transfer to a covered container or jar and refrigerate for 4 weeks.
Strain the sauce twice. First through a fine-mesh sieve, then through a fine-mesh sieve lined with a thin kitchen towel. Transfer to clean bottles or jars. The sauce will keep refrigerated for up to 1 year.