Put chiles in a medium pot and add 1 1/2 qts. water. Soak 24 hours.
With a slotted spoon, transfer chiles to a 1-qt. glass jar. Boil liquid in pot over high heat, uncovered, until reduced to 1 cup, about 30 minutes. Add piloncillo and salt and stir to dissolve. Stir in vinegar.
Pour liquid over chiles. Add remaining ingredients and chill at least 2 weeks.
Serve chiles in a dish and the liquid separately, as a hot sauce.
*Find morita chipotle chiles in Mexican grocery stores. Piloncillo comes in a hard cone and is available at Mexican grocery stores and many supermarkets; grate on the fine holes of a box grater. (For granulated piloncillo, visit ranchogordo.com). Find Mexican oregano in Mexican markets and many supermarkets.
Make ahead: Up to 3 months, chilled.